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MATHEMATICAL  MONOGRAPHS. 

EDITED    BY 

Mansfield  Merriman  and  Robert  S.  Woodward, 

Octavo,  Cloth,  $i.oo  each. 

No.  1.    HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith. 

No.  2.    SYNTHETIC  PROJECTIVE  GEOMETRY. 

By  George  Bruce  Halsted. 

No.  3.    DETERMINANTS. 

By  Laenas  Gifford  Weld. 

No.  4.    HYPERBOLIC   FUNCTIONS. 

By  James  McMahon. 

No.  5.    HARMONIC  FUNCTIONS. 

By  William  E.  Byerly. 

No.  6.    QRASSMANN'S  SPACE  ANALYSIS. 

By  Edward  W.  Hyde. 

No.  7.    PROBABILITY   AND  THEORY   OP   ERRORS. 

By  Robert  S.  Woodward. 

No.  8.    VECTOR  ANALYSIS  AND  QUATERNIONS, 

By  Alexander  Macfarlanb. 

No.  9.    DIFFERENTIAL  EQUATIONS. 

By  William  Woolsey  Johnson. 

No.  10.  THE  SOLUTION  OF  EQUATIONS. 

By  Mansfield  Merriman. 

No.  n.    FUNCTIONS  OF  A  COMPLEX  VARIABLE. 

By  Thomas  S.  Fiske. 

PUBLISHED   BY 

JOHN  WILEY  &  SONS,   NEW  YORK. 
CHAPMAN  &  HALL,  Limited,  LONDON. 


MATHEMATICAL    MONOGRAPHS, 

EDITED    BY 

MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD. 


■    No.  2. 

SYNTHETIC  . 

PROJECTIVE  GEOMETRY, 


BY 

GEORGE    BRUCE    HALSTED, 

Professor  of  Mathematics  in  Kenyon  College. 


FOURTH    EDITION,  ENLARGED. 
FIRST   THOUSAND. 


NEW  YORK: 

JOHN   WILEY   &   SONS. 

London:    CHAPMAN  &  HALL,    Limited. 

1906. 


Hoc 


Copyright,  1896, 

BY 

MANSFIELD  MERRIMAN  and  ROBERT  S.  WOODWARD 

UNDER   THE    TITLE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,  January,  1906. 


1K>BERT  DRUMMOND,   PRINTER.   NBW  YORIC. 


EDITORS'  PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  Hterature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elliptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
Euclidean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  publication  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


OQaQ>^4 


AUTHOR'S  PREFACE. 


Man,  imprisoned  in  a  little  body  with  short-arm  hands  instead 
of  wings,  created  for  his  guidance  a  mole  geometry,  a  tactile 
space,  codified  by  Euclid  in  his  immortal  Elements,  whose  basal 
principle  is  congruence,  measurement. 

Yet  man  is  no  mole.  Infinite  feelers  radiate  from  the  win- 
dows of  his  soul,  whose  wings  touch  the  fixed  stars.  The  angel 
of  light  in  him  created  for  the  guidance  of  eye-life  an  inde- 
pendent system,  a  radiant  geometry,  a  visual  space,  codified  in 
1847  by  a  new  Euclid,  by  the  Erlangen  professor,  Georg  von 
Staudt,  in  his  immortal  Geometrie  der  Lage  published  in  the  quaint 
and  ancient  Niirnberg  of  Albrecht  Diirer. 

Born  on  the  24th  of  January,  1798,  at  Rothenburg  ob  der 
Tauber,  von  Staudt  was  an  aristocrat,  issue  of  the  union  of  two 
of  the  few  regierenden  families  of  the  then  still  free  Reichsstadt, 
which  four  years  later  closed  the  630  years  of  its  renowned  exist- 
ence as  an  independent  republic. 

This  creation  of  a  geometry  of  position  disembarrassed  of  all 
quantity,  wholly  non-metric,  neither  positively  nor  negatively 
quantitative,  resting  exclusively  on  relations  of  situation,  takes 
as  point  of  departure  the  since-famous  quadrilateral  construction. 
To-day  it  must  be  reckoned  with  from  the  abstractest  domains 
of  philosophy  to  the  bread-winning  marts  of  applied  science. 
Thus  Darboux  says  of  it:  "It  seems  to  us  that  under  the  form 
first  given  it  by  von  Staudt,  projective  geometry  must  become 
the  necessary  companion  of  descriptive  geometry,  that  it  is  called 
to  renovate  this  geometry  in  its  spirit,  its  procedures,  its  applica- 
tions." 

Kenyon  College,  Gambier,  Ohio, 
December,  1905. 


CONTENTS. 


Introduction Page  7 

Art.    I.  The  Elements  and  Primal  Forms 8 

2.  Projecting  and  Cutting 10 

3.  Elements  at  Infinity 10 

4.  Correlation  and  Duality 12 

5.  POLYSTIMS  and  POLYGRAMS 12 

6.  Harmonic  Elements 15 

7.  projectivity 18 

8.  Curves  of  the  Second  Degree 20 

9.  Pole  and  Polar 25 

10.  Involution 26 

11.  Projective  Conic  Ranges 29 

12.  Center  and  Diameter 32 

13.  Plane  and  Point  Duality 34 

14.  Ruled  Quadric  Surfaces 36 

15.  Cross-Ratio 42 

16.  homography  and  reciprocation 45 

17.  Transformation.    Pencils  and  Ranges  of  Conics    ....  54 
Index 59 


SYNTHETIC   PROJECTIVE  GEOMETRY. 


Introduction. 

Assumption,  (a)  The  aggregate  of  all  proper  points  on  ai 
straight  line  or  'straight'  is  closed  or  made  compendent  by  one 
point  at  infinity  or  figurative  point. 

(6)  With  regard  to  a  pair  of  different  points  of  those  on  a 
straight  all  remaining  fall  into  two  classes,  such  that  every  point 
belongs  to  one  and  only  one. 

(c)  If  two  points  belong  to  different  classes  with  regard  to- 
a  pair  of  points,  then  also  the  latter  two  belong  to  different  classes 
with  regard  to  the  first  two.  Two  such  point  pairs  are  said 
to  'separate  each  other.' 

(d)  Four  different  points  on  a  straight  can  always  be  par- 
titioned in  one  and  only  one  way  into  two  pairs  separating  each 
other. 

(e)  Such  separation  is  projective,  that  is,  is  carried  on  over 
into  ejects  and  cuts,  using  the  words  in  the  sense  explained  in 
Art.  2. 

Definition.  (/)  The  points  -4,  5,  C,  D  on  a  straight  are  in 
the  sequence  A  BCD  if  ^C  and  BD  are  separated  point  pairs. 
Consequently  this  sequence  is  identical  with  the  following  DABCy 
CDAB,  BCD  Ay  where  each  letter  is  substituted  for  the  one 
following  it  and  the  last  for  the  first.  This  procedure  is  called 
cyclic  permutation.  Each  sequence  again  is  identical  with  the 
outcome  of  its  own  reversal,  giving  DCBAj  CBAD,  BADC, 
ADCB. 

Theorem,  (g)  From  any  two  such  of  the  five  sequences 
ABCDy  ABCEy  ABDE,  ACDE,  BCDE,  as  come  from  dropping 
each  one  of  two  consecutive  elements  of  ABCDEj  the  other 
three  follow. 


o  PROJECTIVE    GEOMETRY. 

Definition,  (h)  The  sequences  A  BCD,  ABCE,  ABDE, 
ACDE,  BCDE  give  the  sequence  ABODE. 

Assumption,  (i)  The  points  on  a  straight  can  be  thought 
in  a  sequence  in  one  sense  or  the  opposite  and  so  that:  I.  If 
any  one  point  A  be  given,  there  is  a  sequence  having  the  chosen 
sense  and  A  as  first  point,  in  vi^hich  i)  of  two  points  B  and  C 
always  one,  say  B,  precedes  the  other  (and  then  C  follows  B)\  2) , 
if  B  precedes  C  and  C  precedes  D,  always  B  precedes  D;  3) 
indefinitely  many  points  follow  B  and  precede  C;  4)  there  is  no 
last  point. 

II.  Both  sequences  having  the  same  first  point  and  opposite 
senses  are  reversals  of  one  another. 

III.  Two  sequences  having  the  same  sense  and  different  first 
points,  say  A  and  B,  follow  one  from  the  other  by  that  cyclic 
interchange  which  brings  A  into  the  place  of  B, 

Art.  1.   The  Elements  and  Primal  Forms. 

1.  A  line  determined  by  two  points  on  it  is  called  a 
*  straight.* 

2.  On  any  two  points  can  be  put  one,  but  only  one,  straight, 
their  '  join.' 

3.  A  surface  determined  by  three  non-costraight  points  on 
it  is  called  a  '  plane.* 

4.  Any  three  points,  not  costraight,  lie  all  on  one  and  only 
one  plane,  their  'junction.' 

5.  If  two  points  lie  on  a  plane,  so  does  their  join. 

6.  The  plane,  the  straight,  and  the  point  are  the  elements 
in  projective  geometry. 

7.  A  straight  is  not  to  be  considered  as  an  aggregate  of 
points.  It  is  a  monad,  an  atom,  a  simple  positional  concept  as 
primal  as  the  point.  It  is  the  *  bearer  *  of  any  points  on  it.  It 
is  the  bearer  of  any  planes  on  it. 

8.  Just  so  the  plane  is  an  element  coeval  with  the  point.  It 
is  the  bearer  of  any  points  on  it,  or  any  straights  on  it. 

9.  A  point  is  the  bearer  of  any  straight  on  it  or  any  plane 
on  it. 

10.  A  point  which  is  on  each  of  two  straights  is  called 
their  *  cross.' 


THE    ELEMENTS    AND    PRIMAL    FORMS.  9 

11.  Planes  all  on  the  same  point,  or  straights  all  with  the 
same  cross,  are  called  *  copunctal.' 

12.  Any  two  planes  lie  both  on  one  and  only  one  straight, 
their  *  meet.' 

13.  Like  points  with  the  same  join,  planes  with  the  same 
meet  are  called  costraight. 

14.  A  plane  and  a  straight  not  on  it  have  one  and  only  one 
point  in  common,  their  'pass.' 

15.  Any  three  planes  not  costraight  are  copunctal  on  one 
and  only  one  point,  their  *  apex.' 

16.  While  these  elements,  namely,  the  plane,  the  straight, 
and  the  point,  retain  their  atomic  character,  they  can  be  united 
into  compound  figures,  of  which  the  primal  class  consists  of 
three  forms,  the  '  range,'  the  '  flat-pencil,'  the  *  axial-pencil.' 

17.  The  aggregate  of  all  points  on  a  straight  is  called  a 
*  point-row,'  or  '  range.'  If  a  point  be  common  to  two  ranges, 
it  is  called  their  '  intersection.' 

18.  A  piece  of  a  range  bounded  by  two  points  is  called  a 
"sect.' 

19.  The  aggregate  of  all  coplanar,  copunctal  straights  is 
called  a  *  flat-pencil.'  The  comrr\on  cross  is  called  the  *  pencil- 
point.'     The  common  plane  is  called  the  'pencil-plane.* 

20.  A  piece  of  a  flat-pencil  bounded  by  two  of  the  straights, 
as  '  sides,'  is  called  an  *  angle.* 

21.  The  aggregate  of  all  planes  on  a  straight  is  called  an 
"axial-pencil,*  or  'axial.*  Their  common  meet,  the  *  axis,*  is 
their  bearer. 

22.  A  piece  of  the  axial  bounded  by  two  of  its  planes,  as 
sides,  is  called  an  '  axial  angle.* 

23.  Angles  are  always  pieces  of  the  figure,  not  rotations. 

24.  No  use  is  made  of  motion.  If  a  moving  point  is  spoken 
of,  it  is  to  be  interpreted  as  the  mind  shifting  its  attention. 

25.  When  there  can  be  no  ambiguity  of  meaning,  a  figure 
in  a  pencil,  though  consisting  only  of  some  single  elements  of 
the  complete  pencil,  may  yet  itself  be  called  a  pencil.  Just  so, 
certain  separate  costraight  points  may  be  called  a  range. 


10  PROJECTIVE    GEOMETRY. 

Art.  2.    Projecting  and  Cutting. 

26',  To  *  project '  from  a  fixed  point  M  (the  *  projection- 
vertex  ')  a  figure,  the  *  original/  composed  of  points  B^  Cy  Dy 
etc.,  and  straights  b,  c,  dy  etc.,  is  to  construct  the  *  projecting 
straights  '  MB,  WCy  MDy  and  the  *  projecting  planes '  'Mb^  Mcy 
Md,  Thus  is  obtained  a  new  figure  composed  of  straights  and 
planes,  all  on  My  and  called  an  *  eject  *  of  the  original. 

27.  To  *  cut '  by  a  fixed  plane  //  (the  *  picture-plane ')  a 
figure,  the  *  subject,'  made  up  of  planes  ^,  y,  d,  etc.,  and 
straights  by  Cy  dy  etc.,  is  to  construct  the  meets  ///?,  ^y,  /xd,  and 

the  passes  jixb,  j^Cy  }xd.  Thus  is  obtained  a  new  figure  com^ 
posed  of  straights  and  points,  all  on  //,  and  called  a  *  cut '  of 
the  subject.  If  the  subject  is  an  eject  of  an  original,  the  cut 
of  the  subject  is  an  *  image  '  of  the  original. 

28.  Axial  projection.  To  project  from  a  fixed  straight  m 
(the  *  projection-axis '),  an  original  composed  of  points  By  C,  Dy. 
etc.,  is  to  construct  the  projecting  planes  mBy  mCy  mD.  Thus 
is  obtained  a  new  figure  composed  of  planes  all  on  the  axis  niy 
and  called  an  '  axial-eject '  of  the  original. 

29.  To  cut  by  a  fixed  straight  m  (to  *  transfix  ')  a  subject 
composed  of  planes  /5,  yy  Sy  etc.,  is  to  construct  the  passes 

mpy  myy  mS,  The  cut  obtained  by  transfixion  is  a  range  ori 
the  *  transversal  *  m. 

30.  Any  two  fixed  primal  figures  are  called  *  projective " 
(7^  when  one  can  be  derived  from  the  other  by  any  finite 
number  of  projectings  and  cuttings. 

Art.  3.    Elements  at  Infinity. 

31.  It  is  assumed  that  for  every  element  in  either  of  the 

three  primal  figures  there  is  always  an  element  in  each  of  the 

others. 

♦Pascal  (1625-62)  and  Desargues  (i 593-1662)  seem  to  have  been  the  first  to 
derive  properties  of  conies  from  the  properties  of  the  circle  by  considering  th*^ 
fact  that  these  curves  lie  in  perspective  on  the  surface  of  the  cone. 


ELEMENTS    AT    INFINITY.  H 

32.  On  each  straight  is  one  and  only  one  point  *  at  infinity,* 
or  *  figurative  *  point.  The  others  are  *  proper  *  points.  Any 
point  going  either  way  (moving  in  either  *  sense  *)  ever  forward 
.on  a  straight  is  at  the  same  time  approaching  and  receding 
ifrom  its  point  at  infinity.  The  straight  is  thus  a  closed  line 
vcompendent  through  its  point  at  infinity. 

33.  *  Parallels '  are  straights  on  a  common  point  at  infinity. 

34.  Two  proper  points  in  it  divide  a  range  into  a  finite  sect 
and  a  sect  through  the  infinite.  Its  figurative  point  and  a 
proper  point  in  it  divide  a  range  into  two  sects  to  the  infinite 
<*  rays'). 

35.  All  the  straights  parallel  to  each  other  on  a  plane  are  on 
the  same  point  at  infinity,  and  so  form  a  flat-pencil  whose  pen- 
cil-point is  figurative.  Such  a  pencil  is  called  a  '  parallel-flat- 
pencil.' 

36.  All  points  at  infinity  on  a  plane  lie  on  one  straight  at 
infinity  or  figurative  straight.*  Its  cross  with  any  proper 
straight  on  the  plane  is  the  point  at  infinity  on  the  proper 
straight. 

37.  Parallel-flat-pencils  on  the  same  plane  have  all  a 
straight  in  common,  namely,  the  straight  at  infinity  on  which 
are  the  figurative  pencil-points  of  all  these  pencils. 

38.  Two  planes  whose  meet  is  a  straight  at  infinity  are 
called  parallel. 

39.  All  the  planes  parallel  to  each  other  are  on  the  same 
figurative  straight,  and  so  form  an  axial  pencil  whose  axis  is  at 
infinity.     Such  an  axial  is  called  a  parallel-axial. 

40.  All  points  at  infinity  and  all  straights  at  infinity  lie  on 
a  plane  at  infinity  or  figurative  plane.  This  plane  at  infinity  is 
•common  to  all  parallel-axials,  since  it  is  on  the  axis  of  each. 

Prob.  I.  From  each  of  the  three  primal  figures  generate  the  other 
two  by  projecting  and  cutting. 

*  This  statement  should  not  be  interpreted  as  descriptive  of  the  nature  of 
infinity.  In  the  Function  Theory  it  is  expedient  to  consider  all  points  in  a 
,plane  at  infinity  as  coincident. 


12     "  PROJECTIVE    GEOMETRY. 

Art.  4.    Correlation  and  Duality. 

41.  Two  figures  are  called  *  correlated  '  when  every  element^ 
of  each  is  paired  with  one  and  only  one  element  of  the  other. 
Correlation  is  a  one-to-one  correspondence  of  elements.  The 
paired  elements  are  called  'mates.' 

42.  Two  figures  correlated  to  a  third  are  correlated  to  each 
other.  For  each  element  of  the  third  has  just  one  mate  in 
each  of  the  others,  and  these  two  are  thus  so  paired  as  to  be 
themselves  mates. 

43.  On  a  plane,  any  theorem  of  configuration  and  deter- 
mination, with  its  proof,  gives  also  a  like  theorem  with  its 
proof,  by  simply  interchanging  point  with  straight,  join  with 
cross,  sect  with  angle.* 

This  correlation  of  points  with  straights  on  a  plane  is 
termed  a  '  principle  of  duality.'  Each  of  two  figures  or  theo- 
rems so  related  is  called  the  '  dual '  of  the  other.f 

Prob.  2.  When  two  coplanar  ranges  m^  and  m'  are  correlated  as 
cuts  of  a  flat-pencil  J/,  show  that  the  figurative  point  P^y  or  Q\  of 
the  one  is  mated,  in  general,  to  a  proper  point  P\  or  Q^  ,  of  the 
other. 

Prob.  3.  Give  the  duals  of  the  following: 

i'.  Two  coplanar  straights  determine  a  flat-pencil  on  their  cross. 
2'.  Two  coplanar  flat-pencils  determine  a  straight,  their  '  concur.* 
3i.   Two  points  bound  two  *  explemental '  sects. 
Prob.  4.  To  draw  a  straight  crossing  three  given  straights,  join 
the  passes  of  two  with  a  plane  on  the  third. 

Art.  5.  PoLYSTiMS  and  Polygrams. 

44j.  A  *  polystim  '  is  a  system  of  44'.  A  '  polygram  '  is  a  system 

n  coplanar  points  ('  dots  '),  with  of  n  coplanar  straights  (*  sides  '), 

all   the    ranges   they   determine  with  all  the  flat-pencils  they  de- 

(*  connectors  ').    Assume  that  no  termine  (*  fans  ').      Assume  that 

three  dots  are  costraight.  no  three  sides  are  copunctal. 

*Culmann's  Graphic  Statics  (Zurich,  1864)  made  extensive  use  of  duah'ty. 
Reye's  Geometrie  der  Lage  (Hannover,  1866)  was  issued  as  a  consequence  of  the 
Graphic  Statics  of  Culmann. 

\  In  Analytic  Geometry  the  principle  of  duality  consists  in  the  interpretation 
of  the  same  equation  in  different  kinds  of  coordinates — point  and  Une  or  point 
and  plane  coordinates. 


POLYSTIMS    AND    POLYGRAMS. 


13 


In  each  dot  intersect  («  —  i) 
connectors,  going  through  the  re- 
maining {n  —  i)  dots.  So  there 
are  n{?i  —  i)/2  connectors. 

45,.  For  n  greater  than  3,  the 
connectors  will  intersect  in 
points  other  than  the  dots.  Such 
intersections  are  called  '  codots.' 

46j.  There  are 
n{n  —  i)(:n—  2){n  —  3)/8codots. 


In  each  side  concur  (n  —  i) 
fans,  going  through  the  remain- 
ing {n  —  1)  sides.  So  there  are 
n{n  —  i)/2  fans. 

45'.  For  n  greater  than  3,  the 
fans  will  concur  in  straights  other 
than  the  sides.  Such  concurs 
are  called  '  diagonals.' 

46'.  There  are 
n{?i  —  i){n  —  2)(«  —  3)/8  diago- 


nals. 
Proof  of  46,.  In  a  polystim  of  n  dots  there  are  rt{n  —  i)/2 
connectors.     These  connectors  intersect  in 

[n{n  —  l)/2][n{n  —  i)/2  —  i]/2  =  n{n  —  i)(«'  —  n—  2)/8 

points  ;  i.e.,  the  number  of  different  combinations  of  n{n  —  i)/2 
things,  two  at  a  time. 

But  some  of  these  intersections  are  dots,  and  the  remaining 
ones  are  codots.  Now  («  —  i)  of  these  connectors  meet  at 
each  dot.  Therefore  each  dot  is  repeated  (n  —  i)  («  —  2)/2 
times;  or  the  number  of  times  the  connectors  intersect  in 
points  not  codots,  i.e.  in  dots,  is  n{fi  —  i){it  —  2)/2. 

Therefore  the  number  of  codots  is 

n{n  —  i){n^  —  n  —  2)/8  —  «(«  —  i){n  —  2)/2 
=  Sjiiyi  —  i)/8][/2'  —  «  —  2  —  4(;/  —  2)] 
—  n{n  —  i)(«  —  2)(«  —  3)/8. 


47,.  A  set  of  n  connectors  may- 
be selected  in  several  ways  so 
that  two  and  only  two  contain 
each  one  of  the  n  dots.  Such 
a  set  of  connectors  is  called  a 
*  complete  set '  of  connectors. 

48,.  There  are  (n  —  i)  I/2 
complete  sets  of  connectors. 


47'.  A  set  of  n  fans  may  be 
selected  in  several  ways  so  that 
two  and  only  two  contain  each 
one  of  the  n  sides.  Such  a  set 
of  fans  is  called  a  *  complete  set  * 
of  fans. 

48'.  There  are  (n  —  i)  !/2 
complete  sets  of  fans. 


Proof  of  48^.  In  a  polystim  of  n  dots  there  are  through  any 
single  dot  (n  —  i)  connectors,  and  hence  (n  —  \)(n  —  2)/2 
pairs  of  connectors.     Consider  one  such  pair,  as  BC  and  BE. 


-14  •  PROJECTIVE    GEOMETRY. 

The  number  of  different  sets  (each  of  «  —  2  connectors) 
from  C  to  E  through  A,  D,  F,  G,  etc.  [there  being  {n  —  3) 
such  dots],  is  {n  —  3) !,  i.e.  the  number  of  permutations  of 
{n  —  3)  things.  Hence  the  number  of  complete  sets  of  con- 
nectors having  the  pair  BC  and  BE  is  («  —  3) !  Therefore  the 
whole  number  of  complete  sets  of  connectors  is 

(»  -  I)(»  -  2)[(«  -  3)  !  ]/2  =  («-!)  !/2. 

49i.  In  any  complete  set  of  49'.  In  any  complete  set  of 
•connectors,  when  n  is  even,  the  fans,  when  n  is  even,  the  first  and 
iirst  and  the  («/2+i)th  are  the  («/2+i)th  are  called  'op- 
called  *  opposite  '.  posite.' 

50j.  A  *  tetrastim  '  is  a  system  50'.  A  *  tetragram  '  is  a  system 
•of  four  dots  with  their  six  con-  of  four  straights  with  their  six 
nectors.  Each  pair  of  opposite  fans.  Each  pair  of  opposite  fans 
•connectors  intersect  in  a  codot.  concur  in  a  diagonal.  These 
These  three  codots  determine  three  diagonals  determine  the 
the  '  codot-tristim  "  of  the  tetra-  *  diagonal-trigram  '  of  the  tetra- 
stim. gram. 

51.  Two  correlated  polystims  whose  paired  dots  and  co- 
dots  have  their  joins  copunctal  are  called  'copolar.' 

52.  Two  correlated  polystims  whose  paired  connectors  in- 
tersect and  have  their  intersections  costraight  are  called 
"*  coaxal.' 

53.  If  two  non-coplanar  tristims  be  copolar,  they  are  coaxal. 
For  since  A  A'  crosses  BB',  therefore  AB  and  A'B'  intersect  on 
"the  meet  of  the  planes  of  the  tristims. 

54.  If  two  non-coplanar  tristims  be  coaxal,  they  are  copolar. 
For  since  AB  intersects  A'B\  these  four  points  are  coplanar. 
The  three  planes  ABA'B\  ACA'C\  BCB'C  are  copunctal. 
Hence  so  are  their  meets  AA\  BB\  CC. 

55.  By  taking  the  angle  between  the  planes  evanescent,  is 
seen  that  coplanar  coaxal  tristims  are  copolar  ;  and  then  by 
reductio  ad  absurdum  that  coplanar  copolar  tristims  are  coaxal. 

56.  If  two  coplnnnr  polystims  are  copolar  and  coaxal  they 
are  said  to  be  *comi:)l(;te  plane  perspectives.*     Their  pole  and 


HARMONIC    ELEMENTS.  15 

axis  are  called  the  *  center  of  perspective*  and  the  *  axis  of 
perspective.' 

57.  If  two  coplanar  tristims  are  copolar  or  coaxal,  they  are 
complete  plane  perspectives. 

58.  If  two  coplanar  polystims  are  images  of  the  same  poly- 
stim  from  different  projection  vertices  F, ,  F, ,  they  are  com- 
plete plane  perspectives.  For  the  joins  of  pairs  of  correlated 
points  are  all  copunctal  (on  the  pass  of  the  straight  F,  F, 
with  the  picture  plane),  and  the  intersections  of  paired  con- 
nectors are  all  costraight  (on  the  meet  of  the  picture  plane 
and  the  plane  of  the  original). 

Prob.  5.  In  a  hexastim  there  are  15  connectors  and  45  codots. 
In  a  hexagram  there  are  15  fans  and  45  diagonals. 

Prob.  6.  If  the  vertices  of  three  coplanar  angles  are  costraight, 
their  sides  make  three  tetragrams  whose  other  diagonals  are  copunc- 
tal by  threes  four  times.     [Prove  and  give  dual.] 

Prob.  7.  The  corresponding  sides  of  any  two  funiculars  of  a 
given  system  of  forces  cross  on  a  straight  parallel  to  the  join  of  the 
poles  of  the  two  funiculars. 

Art.  6.    Harmonic  Elements. 

59.  Fundamental  Theorem. — If  two  correlated  tetrastims 
lie  on  different  planes  whose  meet  is  on  no  one  of  the  eight 
<iots,  and  if  five  connectors  of  the  one  intersect  their  mates, 
then  the  tetrastims  are  coaxal.  For  the  two  pairs  of  tristims 
fixed  by  the  five  pairs  of  intersecting  connectors  being  coaxal 
are  copolar.     Hence  the  sixth  pair  of  connectors  are  coplanar. 

60.  If  the  tetrastims  be  coplanar,  and  if  five  intersections  of 
pairs  of  correlated  connectors  are  costraight,  this  the  coplanar 
case  can  be  made  to  depend  upon  the  other  by  substituting 
for  one  of  the  tetrastims  its  image  on  a  second  plane  meeting 
the  first  on  the  bearer  of  the  five  intersections. 

61.  If  the  axis  m  Isb.  figurative  straight,  the  theorem  reads  : 
If  of  two  correlated  tetrastims  five  pairs  of  mated  connectors 
are  parallel,  so  are  the  remaining  pair. 

62.  Four  costraight  points  are  called  '  harmonic  points,*  or 


16 


PROJECTIVE    GEOMETRY. 


a  '  harmonic  range,'  if  the  first  and  third  are  codots  of  a  tetra- 
stim  while  the  other  two  are  on  the  connectors  through  the 
third  codot. 

63.  By  three  costraight  points  and  their  order  the  fourth 
harmonic  point  is  uniquely  determined.  For  if  the  three  points 


in  order  are  A,  B,  C,  draw  any  two  straights  through  A^  and  a 
third  through  B  to  cross  these  at  K  and  M  respectively.  Join 
CKy  crossing  AM  dit  N,  Join  CM,  crossing  AK  dX  L.  Then  the 
join  LN  crosses  the  straight  ABC,  always  at  the  same  point  D, 
the  fourth  harmonic  to  ^^6' separated  from  B, 

64.  In  projecting  from  a  point  not  coplanar  with  it  a 
tetrastim  defining  a  harmonic  range,  the  four  harmonic  points 
are  projected  by  four  coplanar  straights,  called  'harmonic 
straights'  or  a  'harmonic  flat-pencil.* 

65.  The  four  planes  projecting  harmonic  points  from  an* 
axis  not  coplanar  with  their  bearer  are  called  *  harmonic 
planes,*  or  a  *  harmonic  axial-pencil.' 

66.  Projecting  or  cutting  a  harmonic  primal  figure  gives 
always  again  a  harmonic  primal  figure. 

67.  By  three  elements  of  a  primal  figure,  given  which  is  the 
second,  the  fourth  harmonic  is  completely  determined. 

68.  Defining  harmonic  points  by  the  tetrastim  distinguishes 


HARMONIC    ELEMENTS.  iT 

two  points  made  codots  from  the  other  two.  Yet  it  may  be 
shown  that  the  two  pairs  of  points  play  identically  the  same 
role. 

First,  from  the  definition  of  four  harmonic  points  each  sep- 
arated two  may  be  interchanged  without  the  points  ceasing  to 


be  harmonic  [or,  if  A  BCD  is  a  harmonic  range,  so  is  also 
ADCB,  CBAD,  and  CBAB].  For  the  first  and  third  remain 
codots. 

Second,  to  prove  that  in  a  harmonic  range  the  two  pairs  of 
separated  points  may  be  interchanged  without  the  four  points 
ceasing  to  be  harmonic  [or,  if  ABCD  is  a  harmonic  range 
(and  therefore  ADCB,  CBAD,  and  CDAB),  then  also  is  BADC, 
DABC,  BCDA,  and  DCBA'\  :  Through  the  third  codot  O  draw 
the  joins  AO  and  CO.  These  determine  on  the  connectors 
NK,  KL,  LM,  and  MN  four  new  points,  5,  T,  U,  F,  respec- 
tively. The  tetrastim  KTOS  has  for  two  codots  A  and  C,  and 
has  a  connector  though  B ;  hence  its  remaining  connector  TS 
must  pass  though  D.  In  Hke  manner,  the  connector  UV  of 
the  tetrastim  i^F(9 6^  must  pass  through  D,  and  a  connector 
of  each  of  the  tetrastims  LUG T and  VNSO  through  B.  There- 
fore B  and  D  are  codots  of  a  tetrastim  STUV  with  the  remain- 
ing  connectors,  one  through  Ay  one  through  C. 

69.  The  separated  points  A  and  C  are  called  *  conjugate 
points,'  as  also  are  B  and  D.  Either  two  are  said  to  be  *  har- 
monic conjugates '  with  respect  to  the  other  two. 

Prob.  8.  To  determine  the  join  of  a  given  point  M  with  the  in- 
accessible cross  X  of  two  given  straights  n  and  n'. 


18 


PROJECTIVE    GEOMETRY. 


Through  J/draw  any  two  straights  crossing  natB  and  -5',  and  «' 
at  ^  and  £>\    Join  £>B  and  Z>^B',  crossing  on  A.    Through  A  draw 

any  third  straight  crossing  . 
n  at  ^"  and   «'  at  Z>". 
Join    ^'Z>"   and   Z>'^", 
crossing  at  Z.     Then  ZJ/ 
is  the  join  required. 

Proof.  The  tetrastim 
XBMD  makes  AB'C'D' 
a  harmonic  range,  as 
XB'LD'  does  AB"C"D'\  But  projecting  AB''C"D"  from  X, 
and  cutting  the  eject  by  AB'JD'  gives  a  harmonic  range.  Therefore 
C,  C\  and  X  are  costraight.* 

Prob.  9.  Through  a  given  point  to  draw  with  the  straight-edge 
a  straight  parallel  to  two  given  parallels. 

Prob.  10.  To  determine  the  cross  of  a  given  straight  m  with  the 
inconstructible  join  x  of  two  given  points  N  and  N',  Join  any  two 
points  on  m  withiV" 
and  N\  giving  b 
and  b'  on  iV,  d  and 
d'  on  W.  Join  the 
crosses  db  and  ^'<^' 
by  a.  On  ^  take 
any  third  point  join- 
ing with  N  in  b" 
and  with  N'  in  ^". 
Join  the  crosses  b'd"  and  ^'^  by  /.  Then  Im  is  the  cross  re* 
quired.     [From  Prob.  8,  by  duality.] 

Prob.  II.  Cut  four  coplanar  non-copunctal  straights  in  a  har- 
monic range. 

Prob.  12.  On  a  given  straight  determine  a  point  from  which  the 
ejects  of  three  given  points  form  with  the  given  straight  a  harmonic 
pencil. 

Art.  7.  Projectivity. 
70.  Two  primal  figures  of  three  elements  are  always  pro- 
jective.— If  one  be  a  pencil,  take  its  cut  by  a  transversal.  If 
the  bearers  of  ABC  and  A'B'C  be  not  coplanar,  join  AA\ 
BB' y  CCy  and  cut  these  joins  by  a  transversal,  m.  Then  ABC 
and  A'B'C  are  two  cuts  of  the  axial  mAA',  mBB' ,  mCC , 

*  Numerous  problems  in  Surveying  may  be  solved  by  the  application  of  th« 
preceding  principles,  but  such  application  has  not  been  found  advantageous  Iq 
practice.     See  Gillespie's  Treatise  on  Land  Surveying,  New  York,  1872. 


PROJECTIVITY.  19 

If  the  bearers  are  coplanar,  take  on  the  join  AA^  any  two 
projection  vertices  M  and  M\  Join  MB  and  M'B\  crossing 
at  B''\  join  MC  and  M'C,  crossing  at  C\  Join  B"C'  crossing 
AA'  at  A'\     Then  ABC  and  A'B'C  are  images  of  A"B"C", 

71.  If  any  four  harmonic  elements  are  taken  in  one  of  two 
projective  figures,  the  four  elements  correlated  to  these  are  also 
harmonic.  For  both  ejects  and  cuts  of  harmonic  figures  are 
themselves  harmonic. 

72.  Two  primal  figures  are  projective  if  they  are  so  corre- 
lated that  to  every  four  harmonic  elements  of  the  one  are 
correlated  always  four  harmonic  elements  of  the  other.  For 
the  same  projectings  and  cuttings  which  derive  A' B' C  from 
ABC^AW  give  D^  from  D,  Therefore  A'B'C'D,  is  harmonic. 
But  by  hypothesis  A' B' C D'  is  harmonic.     Therefore  D^  is  U, 

73.  If  two  primal  figures  are  projective,  then  to  every  con- 
secutive order  of  elements  of  the  one  on  a  bearer  corresponds 
a  consecutive  order  of  the  correlated  elements  of  the  other  on 
a  bearer. 

74.  Two  projective  primal  figures  having  three  elements 
self-correlated  are  identical.  For  two  self-correlated  elements 
cannot  bound  an  interval  containing  no  such  element,  since 
they  must  harmonically  separate  one  without  it  from  one 
within. 

75.  Two  ranges  are  called  *  perspective  *  if  cuts  of  the  same 
flat  pencil.  // 

Two  flat  pencils  are  perspective  if  cuts  of  the  same  axial 
pencil,  or  ejects  of  the  same  range.  Two  axials  are  perspective 
if  ejects  of  the  same  flat  pencil. 

A  range  and  a  flat  pencil,  a  range  and  an  axial  pencil,  or  a 
flat  pencil  and  an  axial  are  perspective  if  the  first  is  a  cut  of 
the  second. 

761.  If  two  projective   ranges         76'.  If  two  coplanar  projective 

not  costraight  have  a  self-corre-  flat  pencils  not  copunctal   have 

lated  point  A»  they  are  perspec-  a  self-correlated  straight  tz,  they 

tive.  are  perspective. 


20 


PROJECTIVE    GEOMETRY, 


Let  the  join  of  any  pair  of 
correlated  points  BB'  cross  the 
join  of  any  other  pair  CC  at  V. 

Projecting  the  two  given 
ranges  from  Vy  their  ejects  are 
identical,  since  they  are  projec- 
tive and  have  the  three  straights 
VA,  VBB\  VCC  self-corre- 
lated. 


Let  the  cross  of  any  pair  of 
correlated  straights  bb'  join  the 
cross  of  any  other  pair  cc'  by  m. 

Cutting  the  two  given  flat  pen- 
cils by  niy  their  cuts  are  identical, 
since  they  are  projective  and 
have  the  three  points  may  mbb\ 
mcc'  self-correlated. 


Art.  8.  Curves  of  the  Second  Degree. 


77,.  If  two  coplanar  non- 
copunctal  flat  pencils  are  pro- 
jective but  not  perspective,  the 
crosses  of  correlated  straights 
form  a  *  range  of  the  second  de- 
gree,* or  *  conic  range.' 


77'.  If  two  coplanar  non- 
costraight  ranges  are  projective 
but  not  perspective,  the  joins  of 
correlated  points  form  a  '  pencil 
of  the  second  class,'  or  'conic 
pencil.' 


781.  If  two  copunctual  non- 
-costraight  axial  pencils  are  pro- 
jective but  not  perspective,  the 
meets  of  correlated  planes  form 
a  *  conic  surface  of  the  second 
order,'  or  *cone.' 


78'.  If  two  copunctal  non- 
coplanar  flat  pencils  are  projec- 
tive but  not  perspective,  the 
planes  of  correlated  straights 
form  a  *  pencil  of  planes  of  the 
second  class,'  or  *  cone  of  planes.' 


79.  All  results  obtained  for  the  conic  range  or  the  conic 
pencil  are  interpretable  for  the  cone  or  cone  of  planes,  since 
the  eject  of  a  conic  is  a  cone  and  the  cut  of  a  cone  is  a  conic. 

80'.  On  the  join  a  of  any  pair 
of  correlated  points  A  and  A^  of 


80j.  On  the  cross  A  of  any  pair 
of  correlated  straights  a  and  a^ 


CURVES   OF    THE    SECOND    DEGREE. 


21 


of  the  projective  flat  pencils  V 


and    F,   draw   two    straights    u 
and  u^. 

The  cuts  ABC  and  A.B^C, 
being  projective  and  having  a 
pair  of  correlated  points  Ay  A^ 
coincident,  are  perspective,  both 
being  cuts  of  the  pencil  on  F,, 
the  cross  of  the 
joins  BB^  and 
CC,. 

Any  straight 
<f  of  F,  crossing 
u  at  D,  is  then 
correlated  to 
the  join  of  V^ 
with  the  cross 
Z>j  of  u^  and 
the  join  B>V^. 
Any  d  crosses 
its  d^  so  deter- 
mined, at  P,  a  point  of  the  conic 
range  k. 

8 1  J.  The  pencil-points  F,  Vi 
of  the  generating  pencils  pertain 
to  the  conic,  since  their  join 
FF,  is  crossed  by  the  element 
correlated  to  it  in  either  pencil 
at  its  pencil-point. 


the  projective  ranges  u  and  u^ 
^-  take  two  points  V  and  V^, 
The  ejects  abc  and  a^b^c^ 
being  projective  and  hav- 
ing a  pair  of  correlated 
straights  ^,  a^  coincident, 
are  perspective,  both  be- 
ing ejects  of  the  range  on 
u^y  the  join  of  the  crosses 
bb^  and  cc^. 

Any  point  B>  of  «, 
joined  with  V  by  dy  is 
then  correlated  to  the 
cross  of  u^  with  the  join  d^  of  V^ 
and  the  cross  du^. 

Any  D  joined  to  its  Z>,  so  de- 
termined, gives  p  a  straight  of 
the  conic  pencil  K, 


8i'.  The  bearers  «,  «,  of  the 
generating  ranges  pertain  to  the 
conic,  since  their  cross  uu^  is 
joined  to  the  element  correlated 
to  it  in  either  range  by  its  bearer. 


22 


PROJECTIVE   GEOMETRY. 


82,.  The  Straight  on  F  corre- 
lated to  V^  V  is  called  the  *  tan- 
gent' at  K  Every  other  straight 
on  V  is  its  join  with  a  second 
point  of  the  conic. 

2>T,^,  On  any  straight,  as  Uy  on 
any  point  A  of  the  conic,  its 
second  element  is  its  cross  M 
with  the  join  V^V^. 

84^.  From  the  five  given  points 
VV^  AML^  of  k  construct  a  sixth, 
P,  The  cross  D  oi  u  with  the 
join  VPy  and  the  cross  D^  of  u^ 
with  the  join  V^P  are  costraight 
with  F],.  Therefore*  the  three 
opposite  pairs  in  every  complete 
set  of  connectors  of  a  hexastim 
whose  dots  are  in  a  conic  inter- 
sect in  three  costraight  codots 
whose  bearer  is  called  a  *  Pascal 
straight.' 

This  hexastim  has  sixty  Pascal 
straights,  since  it  has  sixty  com- 
plete sets  of  connectors. 

851.  The  ejects  of  the  points 
of  a  conic  from  any  two  are  pro- 
jective. 

86j.  By  five  of  its  points  a 
conic  is  completely  determined. 

87j.  Instead  of  five  points 
may  be  given  the  two  pencil- 
points  and  three  pairs  of  corre- 
lated straights.  If  one  given 
straight  is  the  join  of  the  pencil- 
points,  then  four  points  and  a 
tangent  at  one  of  them  are  given. 

Thus  by  four  of  its  points  and 
the  tangent  at  one  of  them   a 
*  Pascal,  1640. 


82'.  The  point  on  u  correlated 
to  u^u  is  called  the  *  contact '  on 
u.  Every  other  point  on  u  is  its 
cross  with  a  second  straight  of 
the  conic. 

83'.  On  any  point,  as  F,  on 
any  straight  a  of  the  conic,  its 
second  element  is  its  join  q  with 
the  cross  u^u^* 

84'.  From  the  five  given 
straights  u^  «j,  a^  q,  r^,  of  ^  con- 
struct a  sixth  P>D  ,  or  p.  The 
join  d  oi  V  with  the  cross  upt 
and  the  join  di  of  V^  with  the 
cross  u^p  are  copunctal  with  u^. 
Therefore  f  the  three  opposite 
pairs  in  every  complete  set  of 
fans  of  a  hexagram  whose  sides 
are  in  a  conic  concur  in  three 
copunctal  diagonals  whose  bearer 
is  called  a  *  Brianchon  point.' 

This  hexagram  has  sixty  Brian- 
chon points,  since  it  has  sixty 
complete  sets  of  fans. 

85'.  The  cuts  of  the  straights 
of  a  conic  by  any  two  are  pro- 
jective. 

86'.  By  five  of  its  straights  a 
conic  is  completely  determined. 

87'.  Instead  of  five  straights 
may  be  given  the  two  bearers 
and  three  pairs  of  correlated 
points. 

If  one  given  point  is  the  cross 
of  the  bearers,  then  four  straights 
and  a  contact  point  on  one  of 
them  are  given. 

Thus  by  four  of  its  straights 
and  a  contact-point  on  one  of 
f  Brianchon,  1806. 


CURVES   OF    THE    SECOND    DEGREE. 


33 


conic  is  completely  determined. 

881.  By  three  of  its  points 
and  the  tangents  at  two  of  them 
the  conic  is  completely  deter- 
mined. 

89^.  Interpreting  a  pentastim 
as  a  hexastim  with  two  dots 
coinciding  gives:  In  every  com- 
plete set  of  connectors  of  a  pen- 
tastim whose  dots  are  in  a  conic, 
two  pairs  of  non-consecutive 
connectors  determine  by  their 
two  intersections  a  straight  on 
which  is  the  cross  of  the  fifth 
connector   with   the   tangent   at 


them  a  conic  is  completely  de- 
termined. 

88'.  By  three  of  its  straights 
and  the  contact-points  on  two 
of  them  the  conic  is  completely 
determined. 

89'.  Interpreting  a  pentagram 
as  a  hexagram  with  two  sides 
coinciding  gives:  In  every  com- 
plete set  of  fans  of  a  pentagram 
whose  sides  are  in  a  conic,  two 
pairs  of  non-consecutive  fans 
determine  by  their  two  concurs 
a  point  on  which  is  the  join  of 
the  fifth  fan-point  with  the  con- 
tact-point on  the  opposite  side. 


the  opposite  dot. 

Thence  follows  the  solution  of  the  problems : 


90j.  Given  five  points  of  a 
conic,  to  construct  tangents  at 
the  points,  using  the  ruler  only. 

9 1  J.*  The  hexastim  with  a 
pair  of  opposite  connectors  re- 
placed by  tangents  gives:  The 
int^sections  of  the  two  opposite 
pairs  in  every  complete  set  of 
connectors  of  a  tetrastim  with 
dots  in  a  conic  are  both  costraight 
with  the  crosses  of  the  two  pairs 
of  tangents  at  opposite  dots. 

Or:  A  tetrastim  with  dots  in 
a  conic  has  each  pair  of  codots 
costraight  with  a  pair  of  fan- 
points  of  the  tetragram  of  tan- 
gents at  the  dots. 

The  figure   for    91,   and    that   for  91'  are   identical,   and 
called  Maclaurin's  Configuration.     (See  page  86.) 

92^.  The  tangents  of  a  conic  92'.  The   contact-points  of  9 

range  are  a  conic  pencil.  conic  pencil  are  a  conic  range. 

*  Due  to  Maclaurin,  1748. 


90'.  Given  five  straights  of  a 
conic,  to  find  contact-points  on 
the  straights,  using  the  ruler  only. 

91'.  The  hexagram  with  a  pair 
of  opposite  fans  replaced  by  con- 
tact-points gives:  The  concurs 
of  the  two  opposite  pairs  in  every 
complete  set  of  fans  of  a  tetra- 
gram with  sides  in  a  conic  are 
both  copunctal  with  the  joins  of 
the  two  pairs  of  contact-points 
on  opposite  sides. 

Or:  A  tetragram  with  sides  in  a 
conic  has  each  pair  of  diagonals 
copunctal  with  a  pair  of  con- 
nectors of  the  tetrastim  of  con- 
tacts on  the  sides. 


;24 


PROJECTIVE    GEOMETRY. 


93.  The  points  of  a  conic  range  may  now  be  conceived  as 
all  on  a  curve,  a  ^  conic  curve,'  their  bearer.     The  straights  of 

the  corresponding  conic  pencil, 
tangents  of  this  conic  range,  may 
now  also  be  conceived  as  all  on 
this  same  conic  curve  on  which 
are  their  contact-points.  Conse- 
quently the  conic  curve  is  dual  to 
itself,  and  so  the  principle  of  dual- 
ity on  a  plane  receives  an  impor- 
tant extension. 

94.  It  follows  immediately  from 
their  generation  that  all  conies  are 
closed  curves,  though  they  may 
be  compendent  through  one  or 
two  points  at  infinity.  With  two 
points  at  infinity  the  curve  is  called 
'  hyperbola  ;'  with  one,  *  parabola ;  * 
with  none,  *  ellipse.'  ^ 

95.  If  a   point  has  on  it   tan- 
gents  to    the    curve,    it    is   called 
'without*    the     curve;     if    none, 
*  within  '  the  curve.     The  contact- 
point  on  a  tangent  is  '  on  *  the  curve  ;  all  other  points  on  a  tan- 

*  The  generation  shows  that  a  straight  cuts  the  curves  in  two  points  and 
that  from  any  point  two  tangents  to  the  curves  may  be  drawn.  Hence  the 
curves  are  of  the  second  order  and  of  the  second  class,  that  is  they  are  identical 
with  the  conies  of  analytic  geometry.  Analytically  the  equations  jP-{-XQ  =  o, 
J^'  +  XQ'  —  o,  where  F,  Q,  P',  Q  are  linear  functions  of  point  coordirtates, 
represent  two  projective  pencils,  the  correlated  rays  corresponding  to  the  same 
value  of  X.  Hence  the  locus  of  the  intersection  of  correlated  rays  is  repre- 
sented by  PQ  —  P' Q  ■=  o,  a  second-degree  point  equation.  Projective  ranges 
are  represented  hy  R  -{-  \S  =0,  ^'4-  XS'  =  o,  where  P,  S,  P\  S'  are  linear 
functions  of  line  coordinates.  The  envelope  of  the  joins  of  correlated  points  is 
represented  by  PS'  —  P'  S  =  o,  a  second-degree  line  equation. 

The  projective  generation  of  conies  is  developed  synthetically  in  Steiner's 
Theorie  der  Kegelschnitte,  1866,  and  in  Chasles*  G6om6trie  sup6rieure,  1852. 
For  the  analytic  treatment  see  Clebsch,  Geometrie,  vol.  i,  1876. 


POLE    AND    POLAR.  25 

gent  are  without  the  curve.  Every  straight  in  its  plane  con- 
tains innumerable  points  without  the  curve,  since  the  straight 
-crosses  every  tangent. 

Prob.  13.  Given  four  points  on  a  conic  and  the  tangent  at  one 
of  them,  draw  the  tangent  at  another. 

Prob.  14.  If  the  n  sides  of  a  polygram  rotate  respectively  about 
n  fixed  points  not  costraight,  while  («  — i)  of  a  complete  set  of  fan- 
points  glide  respectively  on  {n  —  \)  fixed  straights,  then  every  remain- 
ing fan-point  describes  a  conic* 

Prob.  15.  In  any  tristim  with  dots  on  a  conic  the  three  crosses 
of  the  connectors  with  the  tangents  at  the  opposite  dots  are 
costraight.f 

Prob.  16.  If  two  given  angles  rotate  about  their  fixed  vertices 
so  that  one  cross  of  their  sides  is  on  a  straight,  either  of  the  other 
three  crosses  describes  a  conic.  J 

Prob.  17.  Construct  a  hyperbola  from  three  given  points,  and 
straights  on  its  figurative  points. 

Art.  9.    Pole  and  Polar. 

96.  Taking  every  tangent  to  a  conic  as  the  dual  to  its  own 
contact-point  fixes  as  dual  to  any  given  point  in  the  plane  one 
particular  straight,  its  *  polar,'  of  which  the  point  is  the 
*  pole.* 

97.  With  reference  to  any  given  conic,  to  construct  the 
polar  of  any  given  point  in  its  plane.  Put  on  the  given  point 
Z  two  secants  crossing  the  curve,  one  at  A  and  D,  the  gther  at 
B  and  C.  The  join  of  the  other  codots  ^and  Fof  ABCD  is 
the  polar  of  Z,  Varying  either  secant,  as  ZBC^  does  not 
change  this  polar,  since  on  it  must  always  be  the  cross  ^S"  of 
the  tangents  at  A  and  D,  and  also  the  point  which  D  and  A 
harmonically  separate  from  Z  (given  by  each  of  the  variable 
tetra^tims  BXCY). 

98.  The  join  of  any  two  codots  of  a  tetrastim  with  dots  on 
a  conic  is  the  polar  of  the  third  codot  with  respect  to  that 

*  Due  to  Braikenridge,  1735. 
f  From  Pascal  ;  dual  from  Brianchon. 

X  Given  by  Newton  in  Principia,  Book  I,  lemma  xxi,  under  the  name  of 
•"the  organic  description  "  of  a  conic. 


26  PROJECTIVE    GEOMETRY. 

conic,  and  either  codot  is  the  pole  of  the  join  of  the  other 
two.  Any  point  is  harmonically  separated  from  its  polar  by 
the  conic. 

99.  To  draw  with  ruler  only  the  tangents  to  a  conic  from 
a  point  without,  join  it  to  the  crosses  of  its  polar  with  the 
conic. 

loOj.    Two   points   are   called  100'.  Two  straights  are  called 

*  conjugate'  with  reference  to  a  *  conjugate'  with  reference  to  a. 

conic  if  one  (and  so  each)  is  on  conic  if  one  (and  so  each)  is  on 

the  polar  of  the  other.  the  pole  of  the  other. 

loij.  All  points  on  a  tangent  loi'.  All  straights  on  a  con- 

are  conjugate  to  its  contact-  tact-point  are  conjugate  to  its. 
point.  tangent. 

io2i.  The   points   of   a  range  102'.  The   straights  of  a  flat, 

are  projective  to  their  conjugates  pencil  are  projective  to  their 
on  its  bearer.  conjugates  on  its  bearer. 

io3j.  With  reference  to  a  given  103'.  With  reference  to  a  given 

conic,      the     *  kerncurve,'      the  conic,  the 'kerncurve,' the  poles. 

polars  of  all  points  on  a  second  of  all  tangents  on  a  second  conic 

conic  make  a  conic  pencil,  whose  make  a  conic  range,  whose  bearer 

bearer    is    the   *  polarcurve  *   of  is  the  *  polarcurve  '  of  the  second 

the  second  conic.  conic. 

Prob.  18.  Either  diagonal  of  a  circumscribed  tetragram  is  the 
polar  of  the  cross  of  the  others. 

Prob.  19.  A  pair  of  tangents  from  any  point  on  a  polar  harmoni- 
cally separate  it  from  its  pole. 

Prob.  20.  A  pair  of  tangents  are  harmonic  conjugates  with  respect 
to  any  pair  of  straights  on  their  cross  which  are  conjugate  with, 
respect  to  the  conic. 

Art.  10.    Involution. 

104.  If  in  a  primal  figure  of  four  elements  (a  *  throw  *)  first 
any  two  be  interchanged,  then  the  other  two,  the  result  is  pro- 
jective to  the  original. 

[That  is,  ABCn  a  BADC  a  CDAB  a  JDCBA.'] 
Let  ABCD  be  a  throw  on  m.     Project  it  from  F.     Cut  this 
eject  by  a  straight  {m')  on  A.    The  cut  is  AB'C'D\      Now 
project  ABCD  from  C.    The  cut  of  this  latter  eject  by  F^  is 


INVOLUTION.  27 

B'.B  VH.    Project  B'B  VH  from  D  and  cut  the  eject  by  m'.    The 
cut  is  B'AD'C'i  which  is  perspective  to  BADC. 


\'-i^A 

^A 

^c 

3 "  \ 

\k 

^^ 

S 
^ 

^i:::^ 

^ 

.e 

\^ 

\ 

105.  Two  projective  primal  figures  of  the  same  kind  of  ele- 
ments and  both  on  the  same  bearer  are  called  '  conjective.* 
When  in  two  conjective  primal  figures  one  particular  element 
has  the  same  mate  to  whichever  figure  it  be  regarded  as  be- 
longing, then  every  element  has  this  property. 

If  A  ABB'  is  projective  to  A'AB'X,  then  by  §  104,  A  ABB' 
is  projective  to  AA'XB\  and  having  three  elements  self-corre- 
lated, they  are  identical. 

106.  Two  conjective  figures  such  that  the  elements  are 
mutually  paired  (*  coupled  ')  form  an  *  Involution.*  For  exam- 
ple, the  points  of  a  range,  and,  on  the  same  bearer,  their  con- 
jugates with  respect  to  a  conic,  form  an  involution.  Every 
eject  and  every  cut  of  an  involution  is  an  involution. 

107.  When  two  ranges  are  projective,  the  point  at  infinity 
of  either  one  is  correlated  to  a  point  of  the  other  called  its 
*  vanishing  point.* 

108.  When  two  conjective  ranges  form  an  involution  the 
two  vanishing  points  coincide  in  a  point  called  the  *  center  '  of 
the  involution. 

109.  If  two  figures  forming  an  involution  have  self-corre- 
lated elements,  these  are  called  the  *  double  *  elements  of  the 
involution.  An  involution  has  at  most  two  double  elements  ; 
for  were  three  self-correlated,  all  would  be  self-correlated. 

no.  If  a  primal  figure  of  four  elements  is  projective  with 
a  second  made  by  interchanging  two  of  these  elements,  they 
harmonically  separate  the  other  two. 

For  project  the  range  ABCD  from  Fand  cut  the  eject  by  a 


28 


PROJECTIVE    GEOMETRY. 


Straight  on  A.     The  cut  AB'CD'  is    projective   to   ABCDy, 

which  by  hypothesis  is  projec- 
tive to  ADCB.  Therefore 
ADCB  is  perspective  to 
AB'CD\  So  vac  is  on  the 
cross  X  of  the  joins  DB'  and 
BD',     So  B  and  D  are  codots 

of  the  tetrastim  VD'XB' ,  while  A  and  C  are  on  the  connectors 

through  C\  the  third  codot. 

111.  If  an  involution  has  two  double  elements  these  sepa- 
rate harmonically  any  two  coupled  elements.  Let  A  and  C  be 
the  double  elements.  Then  ABCB'  is  projective  to  AB'CB  % 
therefore  by  §  no  ABCB'  is  harmonic. 

1 1 2.  An  involution  is  completely  determined  by  two  couples. 
For  the  projective  correspondence  AA'B  .  .  .  7\  A'AB'  ...  is 
completely  determined  by  the  three  given  pairs  of  correlated 
elements,  and  since  among  thern  is  one  couple,  so  are  all  corre- 
lated elements  couples. 

113.  When  there  are  double  elements,  then  the  elements 
of  no  couple  are  separated  by  those  of  another  couple.  In- 
versely, when  the  elements  of  one  couple  separate  those  of 
another,  then  the  elements  of  every  couple  are  separated  by 
those  of  every  other,  and  there  are  no  double  elements. 

114'.  The  three  pairs  of  op- 
posite fan-points  of  a  tetragram 
are  projected  from  any  projec- 
tion-vertex by  three  couples  of 
an  involution  of  straights. 


ii4i.  The  three  pairs  of  op- 
posite connectors  of  a  tetrastim 
are  cut  by  any  transversal  in 
three  couples  of  a  point  involu- 


tion. 


*Due  to  Desargues,  1639. 


PROJECTIVE    CONIC    RANGES.  29" 

Let  QRST  be  a  tetrastim  of  which  the  pairs  of  opposite 
connectors  RT  diVid  QS,  ST  and  QR,  QT  and  RS  arc  cut  by 
any  transversal  respectively  in  A  and  A\  B  and  B\  (7  and  C\ 
From  the  projection-vertex  Q,  the  ranges  ATPR  and  ACA' B' 
are  perspective.  But  A  TPR  and  ABA'C  are  perspective  from 
S.  Therefore  ACA' B'  is  projective  to  ABA'C\  and  therefore 
to  A'C'ABi$  104).  Since  thus^  and  A'  are  coupled,  so  (§  105) 
are  B  and  B\  and  6'  and  6^'. 

115.  To  construct  the  sixth  point  C  oi  an  involution  of 
which  five  points  are  given,  draw  through  C  any  straight,  on 
which  take  any  two  points  Q  and  T.  Join  A  7",  B' Q  crossing 
at  R.  Join  BT,  A'Q  crossing  at  5.  The  join  RS  cuts  the 
bearer  of  the  involution  in  C\ 

Prob.  21.  Find  the  center  O  oi  b.  point  involution  of  which  two 
couples  AA'BB'  are  given. 

Prob.  22.  If  two  points  M  and  N  on  m  are  harmonically  sepa- 
rated by  tmo  pairs  of  opposite  connectors  of  a  tetrastim,  then  so  are 
they  by  the  third  pair. 

Prob.  23.  To  construct  a  conic  which  shall  be  on  three  given 
points,  and  with  regard  to  which  the  couples  of  points  of  an  involu- 
tion on  a  given  straight  shall  be  conjugate  points. 

I 
Art.  11.    Projective  Conic  Ranges.     . 

116.  Four  points  on  a  conic  are  called  harmonic  if  they 
are  projected  from  any  (and  so  every)  fifth  point  on  the  conic 
by  four  harmonic  straights. 

117.  A  conic  and  a  primal  figure  or  two  conies  are  called 
projective  when  so  correlated  that  every  four  harmonic  ele- 
ments of  the  one  correspond  to  four  harmonic  elements  of  the 
other. 

118.  If  a  conic  range  and  a  flat  pencil  are  projective,  and 
every  element  of  the  one  is  on  the  correlated  element  of  the 
other,  they  are  called  perspective.  A  conic  is  projected  from 
every  point  on  it  by  a  flat  pencil  perspective  to  it.  Inversely 
the  pencil-point  of  every  flat  pencil  perspective  to  a  conic  is 
on  the  conic. 


30  PROJECTIVE    GEOMETRY. 

119.  Two  conies  are  projective  if  flat  pencils  respectively- 
perspective  to  them  are  projective.  Therefore  any  three 
elements  in  one  can  be  correlated  to  any  three  elements  in 
the  other,  but  this  completely  pairs  all  the  elements. 

120.  Two  different  conic  ranges  on  the  same  bearer  have 
at  most  two  self-correlated  elements. 

121.  Two  different  coplanar  conic  ranges  with  a  point  V 
in  common  are  projective  if  every  two  points  costraight  with 
V  are  correlated.  For  both  are  then  perspective  to  the  flat 
pencil  on  V.  Every  common  point  other  than  V  is  self-corre- 
lated ;  but  V  only  when  they  have  there  a  common  tangent. 
They  can  have  at  most  three  self-correlated  points. 

122.  If  a  flat  pencil  V  and  conic  range  k  are  coplanar  and 
projective  but  not  perspective,  then  at  most  three  straights  of 
the  pencil  are  on  their  correlated  points  of  the  conic ;  but  at 
least  one. 

For  any  flat  pencil  M  perspective  to  k  is  projective  to  F, 
and  with  it  determines  in  general  a  second  conic  range  which 
must  have  in  common  with  k  every  point  which  lies  on  its 
correlated  straight  of  V,  So  if  more  than  three  straights  of  V 
were  on  their  correlated  points  of  k^  the  conies  would  be  iden- 
tical and  V  perspective  to  k. 

Again,  since  every  conic  is  compendent,  and  so  divides  its 
plane  into  two  severed  pieces,  therefore  the  two  different  conies 
if  they  cross  at  their  common  point  M  must  cross  again,  say 
at  P.  In  this  case  the  straights  VP  and  MP  are  correlated, 
and  so  VP  is  on  the  point  P  correlated  to  it  on  k. 

In  case  they  do  not  cross  at  their  common  point  M,  the 
straight  VM  corresponds  to  the  common  tangent  at  M,  and  so 
to  the  point  M  correlated  to  it  on  k, 

123.  Two  projective  conic  ranges  on  the  same  curve  form 
an  involution  if  a  pair  of  points  are  doubly  correlated.  Besides 
the  couple  AA^,  let  B  and  B^  be  any  other  two  correlated 
points,  so  that  AA^B  corresponds  to  A,AB,.  The  cross  of 
AA,  and  BB^  call  U,  and  its  polar  u.     Project  AA^B  from  B^. 


PROJECTIVE    CONIC    RANGES.  31 

Project  A,AB,  from  B,  '  The  ejects  B,{AA,B)  and  B{A,AB,) 
are  projective,  and  having  the  straight  B^B  (or  BB^)  self-corre- 
lated, so  are  perspective.  The  crosses  of  their  correlated  ele- 
ments are  therefore  costraight.  But  the  cross  of  B^A  with  its 
correlated  straight  BA^  is  known  to  be  on  u,  the  polar  of  £/,  the 


cross  of  AA^  with  BB^,  Likewise  the  cross  of  B^A^  with  BA 
is  on  «.  Therefore  the  point  (7,  correlated  tp  C  is  the  cross 
of  CU  with  the  curve.     So  C  and  C^  are  coupled. 

124.  If  two  conic  ranges  form  an  involution,  the  joins  of 
coupled  points  are  all  copunctal  on  the  '  involutioncenter.' 

125.  Calling  projective  the  conic  pencils  dual  to  projective 
conic  ranges,  if  these  ranges  form  an  involution,  so  do  the 
pencils,  and  the  crosses  of  coupled  tangents  are  all  costraight 
on  the  '  involutionaxis.' 

So  two  conic  pencils  forming  an  involution  are  cut  by  each 
of  their  straights  in  two  ranges  forming  an  involution.  Two 
conic  ranges  forming  an  involution  are  projected  from  each  of 
their  points  in  two  flat  pencils,  forming  an  involution. 

126.  If  the  involutioncenter  lies  without  the  conic  bearer 
of  an  involution,  it  has  two  double  elements  where  it  is  cut  by 
the  involutionaxis. 

127.  To  construct  the  self-correlated  points  of  two  pro- 
jective conic  ranges  on  the  same  conic.^Let  A,  B,  C  be  any 
three  points  of  ^,  and  ^,,  B^^  (7,  their  correlated  points  of  k^. 
The  projective  flat  pencils  A{A,B^C,)  and  A^ABC)  have  AA, 
self-corresponding,  hence  they  are  perspective  to  a  range  on 
the  join  u  of  the  cross  of  AB^  and  A^B  with  the  cross  of  AC^ 


32  PROJECTIVE    GEOMETRY. 

and  A^C.    The  crosses  of  the  conic  and  this  join  u  are  the 
self-correlated  points  of  k  and  k^. 

128.  If  the  dots  of  a  tetrastim  are  on  a  conic,  the  six  points 
where  a  straight  not  on  a  dot  cuts  the  conic  and  two  pairs  of 
opposite  connectors  form  an  involution. 

For  the  two  flat  pencils  in  which  the  two  crosses  of  m 
with  the  conic,  P,  P,,  and  two  opposite  dots  R,  T,  are  pro- 
jected from  the  other  two  dots  Q,  S,  are  projective,  and  con- 
sequently so  are  the  cuts  of  these  flat  pencils  by  m;  that  is, 
PBP.A  A  PA,P,B,,  But  PA,P,B,  a  P,B,PA,.  Therefore 
PBP,A  A  P,B,PA,. 

i29i.  Conies  on  which  are  the  129'.  Copunctal   tangents    to 

dots  of  a  tetrastim  are  cut  by  a  conies  on  which  are  the  sides  of 

transversal  in  points  of  an  involu-  a  tetragram  form  an  involution, 

tion.     At  its  double  points  the  The  double  straights  touch  two 

transversal  is  tangent  to  two  of  of   those   conies   at  the   pencil- 

those  conies.  point. 

Prob.  24.  The  pairs  of  points  in  which  a  conic  is  cut  by  the 
straights  of  a  pencil  whose  pencil-point  is  not  on  the  conic  form  an 
involution. 

Art.  12.    Center  and  Diameter. 

130.  The  harmonic  conjugate  of  a  point  at  infinity  with 
respect  to  the  end  points  of  a  finite  sect  is  the  *  center  *  of  that 
sect. 

131.  The  pole  of  a  straight  at  infinity  with  respect  to  a 
certain  conic  is  the  *  center  *  of  the  conic. 

132.  The  polar  of  any  figurative  point  is  on  the  centre  of 
the  conic,  and  is  called  a  '  diameter.* 

133.  If  a  straight  crosses  a  conic  the  sect  between  the 
crosses  is  called  a  *  chord.' 

The  center  of  a  €6hT^  is  the  center  of  all  chords  on  it. 

134.  The  centers  of  chords  on  straights  conjugate  to  a 
diameter  are  all  on  the  diameter. 

135.  Two  diameters  are  conjugate  when  each  is  the  polar 
of  the  figurative  point  on  the  other. 


CENTER    AND    DIAMETER,  33 

136.  The  tangents  at  the  crosses  of  a  straight  with  a  conic 
cross  on  the  diameter  which  is  a  conjugate  to  that  straight. 

137.  The  joins  of  any  point  on  the  conic  to  the  crosses  of  a 
diameter  with  the  conic  are  parallel  to  two  conjugate  diameters. 

138.  Of  two  conjugate  diameters,  each  is  on  the  centers  of 
the  chords  parallel  to  the  other ;  and  if  one  crosses  the  conic, 
the  tangents  at  its  crosses  are  parallel  to  the  other  diameter. 

139.  The  center  of  an  ellipse  is  within  it,  for  its  polar  does 
not  meet  the  curve,  and  so  there  are  no  tangents  from  it  to  the 
curve.  The  centre  of  a  parabola  is  the  contact  point  of  the 
figurative  straight.  The  centre  of  a  hyperbola  lies  without  the 
curve,  since  the  figurative  straight  crosses  the  curve.  The  tan- 
gents from  the  center  to  the  hyperbola  are  called  '  asymptotes/ 
Their  contact-points  are  the  two  points  at  infinity  on  the 
curve. 

140.  If  a  diameter  which  cuts  the  curve  be  given,  the  tan- 
gents at  its  crosses  can  be  constructed  with  ruler  only,  and  so 
however  many  chords  on  straights  conjugate  to  the  diameter. 

141.  Every  flat  pencil  is  an  involution  of  conjugates  with 
respect  to  a  given  conic.  Hence  the  pairs  of  conjugate  diam- 
eters of  a  conic  form  an  involution. 

If  the  conic  is  a  hyperbola,  the  asymptotes  are  the  double 
straights  of  the  involution.  Hence  any  two  conjugate  diam- 
eters of  a  hyperbola  are  harmonically  separated  by  the  asymp- 
totes ;  and  since  the  hyperbola  lies  wholly  in  one  of  the  twa 
explemental  angles  made  by  the  asymptotes,  one  diameter 
cuts  the  curve,  the  other  does  not. 

142.  Any  one  pair  of  conjugate  diameters  of  an  ellipse  is 
always  separated  by  any  other  pair.  Any  one  pair  of  conjugate 
diameters  of  a  hyperbola  is  never  separated  by  any  other  pair. 

143.  If  a  tangent  to  a  hyperbola  cuts  the  asymptotes  at  A 
and  C  then  the  contact-point  B  is  the  center  of  the  sect  AC, 
since  the  tangent  cuts  the  harmonic  pencil  made  by  the  diame- 
ter through  B^  the  conjugate  diameter  and  the  asymptotes,  in 
the  harmonic  range  ABCD  where  D  is  at  infinity.     Just  so  the 


34  PROJECTIVE    GEOMETRY. 

center  of  any  chord  is  the  center  of  the  costraight  sect  bounded 
by  the  asymptotes. 

144.  If  a  point  is  the  center  of  two  chords  it  is  the  center 
of  the  conic,  for  its  polar  is  the  figurative  straight. 

145.  As  many  points  as  desired  of  a  conic  may  be  con- 
structed by  the  ruler  alone. 

With  the  aid  of  one  fixed  conic  all  problems  solvable  by 
ruler  and  compasses  can  be  solved  by  ruler  alone,  that  is,  by 
pure  projective  geometry.  For  example  :  Of  two  projective 
primal  figures  (say  ranges)  on  the  same  bearer,  given  three 
pairs  of  correlated  elements  to  find  the  self-corresponding  ele- 
ments, if  there  be  any.  Project  the  two  ranges  from  any  point 
V  of  the  given  conic.  These  ejects  are  cut  by  the  conic  in 
projective  conic  ranges.  Of  these  determine  the  self-correlated 
points  by  §  127. 

Project  these  from  F.  The  ejects  cut  the  bearer  of  the 
original  ranges  in  the  required  self-correlated  points. 

Prob.  25.  Find  the  crosses  of  a  straight  with  a  conic  given  only 
by  five  points. 

Prob.  26.  Given  a  conic  and  its  center,  find  a  point  B  such  that 
for  two  given  points  A,  C,  the  center  of  the  sect  AB  shall  be  C. 

Prob.  27.  The  join  of  the  other  extremities  of  two  coinitial  sects 
is  parallel  to  the  join  of  their  centers. 

Prob.  28.  In  an  ellipse  let  A  and  B  be  crosses  of  conjugate  diam- 
eters CA,  CB  with  the  curve.  Through  A'  the  cross  of  the  diameter 
conjugate  to  CA  with  the  curve  draw  a  parallel  to  the  join  AB.  Let 
it  cut  the  curve  again  at  B'.  Then  CB'  is  the  diameter  conjugate 
to  CB. 

Art.  13.    Plane  and  Point  Duality. 

1461.  On  a  plane  are  00' points,         146'.  On  a  point  are  00' planes, 

a  *  point-field.'  a  *  plan  e-sh  eaf . ' 

i47i.  The  00*  planes  of  a  sin-         147'.  The  00^  points  of  a  sin- 

:gle  axial  pencil  have  on  them  all  gle  range  have  on  them  all  the 

the   points    of    point-space;    so  planes  of  plane-space;    so   there 

•ihere  are  just  00^  points.  are  just  00'  planes. 

Point-space  is  tridimensional.  Plane-space  is  tridimensional. 


PLANE    AND    POINT    DUALITY. 


35 


148.  With  the  straight  as  element,  space  is  of  four  dimen- 
sions. 


On  a  plane  are  00'  straights, 
a  *  straight-field.' 

On  a  straight  are  00'  planes, 
and  so  00^  straights. 

On  each  of  the  00'  points  on 
a  plane  are  the  00'  straights  of  a 
straight-sheaf;  so  there  are  just 
00*  straights. 

i49j.  Two  planes  determine  a 
straight,  their  meet. 

150,.  Two  planes  determine  an 
axial-pencil  on  their  meet. 

151,.  Two  bounding  planes 
determine  an  axial  angle. 

152,.  A  plane  and  a  straight 
not  on  it  determine  a  point,  their 
pass. 

153,.  An  axial  pencil  and  a 
plane  not  on  its  bearer  deter- 
mine a  flat  pencil. 

i54i.  Three  planes  determine 
a  point,  their  apex. 

i55i.  Three  planes  determine 
a  plane-sheaf. 

156,.  Two  coplanar  straights 
are  copunctal. 


On  a  point  are  00^  straights,  a 
*  straight-sheaf.' 

On  a  straight  are  00*  points,, 
and  so  00^  straights. 

On  each  of  the  00'  planes  on 
a  point  are  the  00'  straights  of  a 
straight-field;  so  there  are  just 
00*  straights. 

149'.  Two  points  determine  a 
straight,  their  join. 

150'.  Two  points  determine  a 
range  on  their  join. 

151'.  Two  bounding  points 
determine  a  sect. 

152'.  A  point  and  a  straight 
not  on  it  determine  a  plane. 


A  range  and  a  point  not 
bearer   determine   a   flat 


153 
on  its 
pencil. 

154'.  Three   points  determine 
a  plane,  their  junction. 

155'.  Three  points   determine 
a  point-field. 

156'.  Two  copunctal  straights 
are  coplanar. 

157.  Any  figure,  or  the  proof  of  any  theorem  of  configu- 
ration and  determination,  gives  a  dual  figure  or  proves  a  dual 
theorem  by  simply  interchanging  point  with  plane.  Thus  all 
the  pure  projective  geometry  on  a  plane  may  be  read  as  geom- 
etry on  a  point. 

Prob.  29.  If  of  straights  copunctal  in  pairs  not  all  are  copunctal^ 

then  all  are  coplanar. 

Prob.  30.  On  a  given  point  put  a  straight  to  cut  two  given  straights, 
Prob.  31.  If  two  triplets  of  planes  oc^y^  a'ft'y'  are  such  that 

the  meets  Py  and  p'y\  ya  and  y'a\  aft  and  «'/?'  lie  on  three 

planes  «",  /?",  ;/"  which  are  costraight,  then  the  meets  aa\  ftft\ 

yy*  are  coplanar. 


36  PROJECTIVE    GEOMETRY. 

Prob.  32.  Describe  the  figures  in  space  dual  to  the  polystim  and 
the  polygram. 

Art.  14.    Ruled  Quadric  Surfaces. 

158.  The  joins  of  the  correlated  points  of  two  projective 
ranges  whose  bearers  are  not  coplanar  form  a  *  ruled  system ' 
of  straights  no  two  coplanar.  For  were  two  coplanar,  then 
two  points  on  the  bearer  m  and  two  on  the  bearer  ;;^,  would 
all  four  be  on  this  plane,  and  so  m  and  7n^  coplanar,  contrary 
to  hypothesis. 

159.  Let  the  straights  7i,  n^,  n^  be  any  three  of  the  elements 
of  a  ruled  system,  and  N^  any  point  on  n^.  Put  a  plane  on  N^ 
and  the  straight  ;^.,  and  let  its  pass  with  n  be  called  N.  The 
straight  iWV,  cuts  n^  n^^  n^  all  three.  Projecting  the  generating 
ranges  of  the  ruled  system  (on  the  bearers  m  and  m^  from  the 
straight  NN^  (or  m^  as  axis  produces  two  projective  axial 
pencils,  which  having  three  planes  m^n,  m^n^,  m^n^  self-corre- 
sponding, are  identical.  Therefore  every  pair  of  correlated 
points  of  the  ranges  on  m  and  m^  is  coplanar  with  m^ ;  that 
is,  m^  cuts  every  element  of  the  ruled  system. 

By  varying  the  point  N^  00*  straights  are  obtained,  all  cutting 
all  the  00*  straights  of  the  original  ruled  system  and  making 
on  every  two  projective  ranges.  Of  the  straights  so  obtained 
no  two  cross,  for  that  would  make  two  of  the  first  ruled  system 
coplanar. 

Either  of  these  two  systems  may  be  considered  as  generating 
a  *  ruled  surface,*  which  is  the  bearer  of  both.  Each  of  the 
two  systems  is  completely  determined  by  any  three  straights 
of  the  other,  and  therefore  so  is  the  ruled  surface  also.  From 
the  construction  follows  that  the  straights  of  either  ruled 
system  cut  all  the  straights  of  the  other  in  projective  ranges. 
So  any  two  straights  of  either  system  may  be  considered  as 
bearers  of  projective  ranges  generating  the  other  system,  or 
indeed  the  ruled  surface. 

160.  On  each  point  of  this  ruled  surface  are  two  and  only 
two  straights  lying  wholly  in  the  surface  (one  in  each  ruled 


RULED    QUADRIC    SURFACES.  37 

system).     So  a  plane  on  one  straight  of  the  ruled  surface  is 

also   on   another  straight  of  this 

surface. 

i6i.  If  in  the  two  generating 
projective  ranges  the  point  at 
infinity  of  one  is  correlated  to  the 
point  at  infinity  of  the  other,  the 
ruled  surface  is  called  a  *  hyper- 
bolic-paraboloid.' 

The   join   of   these  figurative 
points  is  on  the  figurative  plane. 
Therefore   the   plane    at  infinity 
cuts  the  surface  in  a  straight  and  so  has  a  second  straight  in 
common  with  the  ruled  surface. 

That  a  hyperbolic-paraboloid  has  two  straights  in  common 
with  the  plane  at  infinity  may  also  be  proved  as  follows: 

Call  the  bearers  of  the  generating  ranges  m  and  m^,  and  let 
«,  n^  be  any  two  elements,  and /the  element  at  infinity.  By 
§  159  the  ruled  surface  may  be  considered  as  generated  by  the 
straights  on  the  three  elements  w,  «, ,  /.  But  all  these  straights 
must  be  parallel  to  the  same  plane,  namely,  to  any  plane  on  /. 
On /and  each  one  of  these  straights  put  a  plane  ;  these  planes 
make  a  parallel-axial-pencil,  and  cut  any  two  of  the  original 
elements  in  projective  ranges  with  the  figurative  points  corre- 
lated. '  Therefore  the  figurative  straight  joining  the  figurative 
points  of  n  and  «,  is  wholly  on  the  ruled  surface. 

162.  From  §  161  follows  that  all  straights  pertaining  to  the 
same  ruled  system  on  a  hyperbolic-paraboloid  are  parallel  to 
the  same  plane.  Such  planes  are  called  *  asymptote-planes.* 
A  hyperbolic-paraboloid  is  completely  determined  by  two  non- 
coplanar  straights  and  an  asymptote-plane  cutting  them.  To 
get  an  element  cut  the  two  given  straights  by  any  plane  par- 
allel to  the  asymptote-plane,  and  join  the  meets. 

163.  Three  non-crossing  straights,  all  parallel  to  the  same 
plane,  completely  determine  a  hyperbolic-paraboloid.  Let  ;;/, 
^„  w,  be  the  given  straights.     The  passes  of  planes  on   m^ 


38  PROJECTIVE    GEOMETRY. 

with  m  and  m^  are  projective  ranges  whose  joins  are  a  ruled 
system. 

But  from  the  hypothesis  one  of  these  planes  is  parallel  to 
both  m  and  m. .  Therefore  their  points  at  infinity  are  corre- 
lated and  the  ruled  surface  is  a  hyperboHc-paraboloid. 

164.  If  two  non-coplanar  projective  ranges  be  each  axially 
projected  from  the  bearer  of  the  other,  two  projective  axial 
pencils  are  formed,  with  those  planes  correlated  on  which  are 
the  correlated  points  of  the  ranges.  If  A^  A^  be  correlated 
points,  then  the  straight  AA^  is  the  meet  of  correlated  planes. 
Thus  two  projective  axial  pencils  with  axes  not  coplanar  gen- 
erate a  ruled  system.  If  the  whole  figure  be  cut  by  a  plane, 
this  will  cut  these  axial  pencils  in  two  projective  flat  pencils, 
and  the  conic  generated  by  these  will  be  the  cut  of  the  ruled 
surface.  So  every  plane  cuts  it  in  a  conic  or  a  pair  of  straights. 
Hence  no  straight  not  wholly  on  the  surface  can  cut  it  in  more 
than  two  points.  The  surface  is  therefore  of  the  second  degree 
(quadric). 

If  the  plane  at  infinity  cuts  the  ruled  surface  in  a  pair  of 
straights,  it  is  a  hyperbolic-paraboloid.     If  not,  it  is  called  a 

*  hyperboloid  of  one  nappe,'  a  fig- 
ure of  which  is  here  shown. 

164!^.  Copunctal  straights  par- 
allel to  the  generating  elements  of 
a  hyperboloid  of  one  nappe  are  on 
a  cone.  Copunctal  straights  par- 
allel to  the  generating  elements  of 
a  hyperbolic-paraboloid  are  on  a 
system  of  two  planes. 

For  the  figurative  plane  cuts 
the  hyperboloid  of  one  nappe  in  a 
conic  curve,  but  cuts  the  hyper- 
bolic-paraboloid in  two  straights; 
and  each  of  the  copunctal  straights 
goes  to  a  point  of  the  figurative  cut. 

165.  Each  straight  in  one  ruled  system  of  a  hyperboloid  of 


RULED    QUADRIC    SURFACES.  39" 

one  nappe  is  parallel  to  one,  but  only  to  one,  straight  in  the 
other  ruled  system.  Of  the  straights  on  a  hyperbolic-parabo- 
loid no  two  are  parallel.  Let  n  and  n^ ,  any  two  elements  of 
one  ruled  system,  be  the  bearers  of  the  generating  ranges  R 
and  R^,  If  V  is  the  vanishing  point  of  R,  then  the  straight  on 
V  parallel  to  n^  is  an  element  of  the  other  ruled  system.  But 
for  the  hyperbolic-paraboloid  V\s  itself  a  figurative  point. 

1 66.  Any  straight  of  one  ruled  system  on  a  ruled  surface  is 
called  a  *  guide-straight  *  of  the  other  ruled  system. 

1671.  A  ruled  system  is  cut  by  167'.  A  ruled  system  is   pro- 

any  two  of  its  guide-straights  in  jected  from  any  two  of  its  guide- 
projective  ranges.  straights  in  projective  axial  pen- 

cils. 

For  if  7n,  ;«,,  m^  be  any  three  guide-straights  of  the  ruled 
system,  the  planes  on  w,  cut  m  and  w,  in  projective  ranges  the 
joins  of  whose  correlated  points  are  the  elements  of  the  ruled 
system.  Again,  if  the  points  on  w,  be  projected  axially  from 
m  and  ;;/„  the  meets  of  the  planes  so  correlated  are  the  ele- 
ments of  the  ruled  system. 

168.  Four  straights  of  a  ruled  system  are  called  harmonic 
straights  if  they  are  cut  in  four  harmonic  points  by  one  (and  so 
by  every)  guide-straight.  By, three  straights,  no  two  coplanar, 
a  fourth  harmonic  is  determined  lying  in  a  ruled  system  with 
the  given  three  and  on  a  fourth  harmonic  point  to  any  three 
costraight  points  of  the  given  three. 

169.  A  plane  cutting  the  ruled  surface  in  a  straight  m  of  one- 
ruled  system  and  consequently  also  in  a  straight  n  of  the  other 
ruled  system  has  in  common  with  the  surface  no  point  not  on 
one  of  these  straights.  For  any  straight  from  such  a  point 
cutting  both  these  straights  would  lie  wholly  on  the  ruled  sur- 
face ;  and  so  therefore  would  their  whole  plane,  which  is  im- 
possible. Any  third  straight  coplanar  with  m  and  n  on  their 
cross  has  no  second  point  in  common  with  the  surface  and  so 
is  a  tangent,  and  the  plane  of  m  and  n  is  called  tangent  at  their 
cross,  the  point  mn. 


40  PROJECTIVE    GEOMETRY. 

The  number  of  planes  tangent  to  the  ruled  surface  and  on 
a  given  straight  equals  the  number  of  points  the  straight  has 
in  common  with  the  ruled  surface,  that  is  two ;  so  the  ruled 
surface  is  of  the  second  class. 

170.  Project  the  two  generating  ranges  of  a  ruled  system 
from  any  projection-vertex  V  not  on  it.  The  eject  consists  of 
two  copunctal  projective  flat  pencils.  The  plane  of  any  two 
correlated  straights  is  on  an  element  of  the  ruled  system.  All 
such  planes  form  a  cone  of  planes. 

The  points  of  contact  of  these  planes  with  the  ruled  surface 
are  a  conic  range.  The  planes  tangent  to  a  ruled  surface  at 
the  points  on  its  cut  with  a  plane  form  a  cone  of  planes. 

171.  The  cut  of  a  hyperbolic-paraboloid  by  a  plane  not  on 
an  element  has  on  it  the  passes  of  the  plane  with  the  two  figu- 
rative elements,  and  so  is  a  hyperbola  except  when  their  cross 
is  on  the  plane,  in  which  case  it  is  a  parabola.  The  figurative 
plane  is  a  tangent  plane. 

172.  The  planes  tangent  at  the  figurative  points  of  a  hyper- 
boloid  of  one  nappe  are  all  proper  planes,  copunctal  and  form- 
ing a  cone  of  planes  tangent  to  the  *  asymptote-cone  '  of  the 
hyperboloid.  Each  element  to  the  asymptote-cone  is  parallel 
to  one  element  of  each  ruled  system. 

Any  plane  not  on  an  element  of  the  hyperboloid  of  one 
nappe  cuts  it  in  a  hyperbola,  parabola,  or  ellipse,  according  as 
it  is  parallel  to  two  elements,  one,  or  no  element  of  the  asymp- 
tote-cone, that  is,  according  as  it  has  in  common  with  the  figu- 
rative conic  on  the  hyperboloid  two  points,  one,  or  no  point. 

173.  If  an  axial  pencil  and  a  ruled  system  are  projective, 
they  generate  in  general  a  *  twisted  cubic  curve,*  which  any 
plane  cuts  in  one  point  at  least  and  three  at  most.  For  a 
plane  cuts  the  ruled  system  in  a  conic  range  perspective  to  it, 
of  which  in  general  three  points  at  most  lie  on  the  correspond- 
ing planes  of  the  pencil. 

174.  The  ruled  quadric  surface  is  the  only  surface  doubly 


RULED    QUADRIC    SURFACES.  41 

ruled.     The  figure  of  two  so  united  ruled  systems  is  one  of  the 
most  noteworthy  discovered  by  the  modern  geometry.* 

175.  To  find  the  straights  crossing  four  given  straights. — 
Let  «„  «„  u^,  u^  be  the  given  straights.  Projecting  the  range 
R^  on  «j  from  the'  axes  ti^  and  u^  gives  two  axial  pencils,  each 
perspective  to  /?,,  and  consequently  projective.  The  meets  of 
their  correlated  planes  are  all  the  oo'  straights  on  u^,  u^,  u^, 
and  form  a  ruled  system  of  which  «,,  u^,  ti^  are  guide-straights. 
The  two  projective  axial-pencils  cut  the  fourth  straight  u^  in 
two  *  conjective  '  ranges.  [Two  projective  primal  figures  of  the 
same  kind  and  on  the  same  bearer  are  called  conjective.]  If 
now  a  straight  ni  of  the  ruled  system  crosses  2/^,  then  the  two 
correlated  planes  of  which  this  straight  m  is  the  meet  must  cut 
u^  in  the  same  point,  which  consequently  is  a  self-correspond- 
ing point  of  the  two  conjective  ranges.  Since  there  are  two 
such  (the  points  common  to  u^  and  the  ruled  surface),  so  there 
are  two  straights  (real  or  conjugate  imaginary)  crossing  four 
given  straights.  Their  construction  is  shown  to  depend  on 
•that  for  the  two  self-correlated  points  of  two  conjective  ranges. 

This  important  problem  in  the  four-dimensional  space  of 
straights,  *  what  is  common  to  four  straights  ? '  is  the  analogue 
of  the  problem  in  the  space  of  points,  '  what  is  common  to 
three  points?*  and  its  dual  in  the  space  of  planes,  *  what  is 
common  to  three  planes?  * 

It  shows  not  only  their  fundamental  diversity,  but  also,  as 
"compared  to  points-geometry  and  planes-geometry,  the  inher- 
ently quadratic  character  of  straights-geometry. 

Prob.  33.  Find  the  straights  cutting  two  given  straights  and 
parallel  to  a  third. 

Prob.  34.  Three  diagonals  of  a  skew  hexagram  whose  six  sides 
are  on  a  ruled  surface  are  copunctal. 

Prob.  35.  If  a  flat  pencil  and  a  range  not  on  parallel  planes  are 
projective,  then  straights  on  the  points  of  the  range  parallel  to  the 
correlated  straights  of  the  pencil  form  one  ruled  system  of  a  hyper- 
bolic-paraboloid. 

*See  Monge,  Journal  de  Tfecole  poly  technique,  Vol.  I. 


42  PROJECTIVE    GEOMETRY. 

Prob.  36.  What  is  the  locus  of  a  point  harmonically  separated 
from  a  given  point  by  a  ruled  surface  ? 

Art.  15.    Cross-Ratio. 

176.  Lindemann  has  shown  how  every  one  number,  whether 
integer,  fraction,  or  irrational,  -f-  or  — ,  may  be  correlated  to 
one  point  of  a  straight,  without  making  any  use  of  measure- 
ment, without  any  comparison  of  sects  by  application  of  a  unit 
sect.*  He  gets  an  analytic  definition  of  the  *  cross-ratio  '  of 
four  copunctal  straights.  Then  this  expression  is  applied  to 
four  costraight  points.  Then  is  deduced  that  the  number  pre- 
viously attached  to  a  point  on  a  straight  is  the  same  as  the 
cross-ratio  of  that  point  with  three  fixed  points  of  the  straight.. 
Thus  analytic  geometry  and  metric  geometry  may  be  founded 
without  using  ratio  in  its  old  sense,  involving  measurement. 
Thus  also  the  non-Euclidean  geometries,  that  of  Bolyai-Loba- 
ch^vski  in  which  the  straight  has  two  points  at  infinity,  and 
that  of  Riemann  in  which  the  straight  has  no  point  at  infinity, 
may  be  treated  together  with  the  limiting  case  of  each  between 
them,  the  Euclidean  geometry,  wherein  the  straight  has  one 
but  only  one  point  at  infinity. 

Relinquishing  for  brevity  this  pure  projective  standpoint 
and  reverting  to  the  old  metric  usages  where  an  angle  is  an  in- 
clination, a  sect  is  a  piece  of  a  straight,  and  any  ratio  is  a 
number;  distinguishing  the  sect  AC  from  CA  as  of  opposite 
*  sense,'  so  that  AC=  -  CA,  the  ratio  {AC/BCViAD/BD']  is 
called  the  cross-ratio  of  the  range  ABCD  and  is  written  \ABCD'\ 
where  A  and  B,  called  conjugate  points  of  the  cross-ratio,  may 
be  looked  upon  as  the  extremities  of  a  sect  divided  internally 
or  externally  by  C  and  again  by  D,\ 

*Von  Staudt  in  Beitrage  zur  Geometrie  der  Lage,  1856-60,  determines  the 
projective  definition  of  number,  and  thus  makes  the  metric  geometry  a  conse- 
quence of  projective  geometry. 

f  The  fundamental  property  of  cross-ratio  is  stated  in  the  Mathematical  Col- 
lections of  Pappus,  about  370  a.d.  The  cross-ratio  is  the  basis  of  Poncelet's 
Traitfe  des  propri6t6s  projectives,  1822,  which  distinguishes  sharply  the  projec- 
tive and  metric  properties  of  curves. 


CROSS-RATIO.  43 

177.  If  on  ABCD  respectively  be  the  straights  abed  co- 
punctal  on  V,  then  A  C/BC=  A  A  VC/AB  VC 

or  A  C/BC  =  iA  V.  VC  sin  {ac)/iB  V.  VC  sin  {be), 
AD/BD  =  A  A  VD/AB  VD 

=  iA  V,  VD  sin  {aa)/iB  V.  VD  sin  {bd). 

Therefore       [ABCD']  =  [sin  (ac)/sm  {be)y[sm  ad/sin  {bd)] . 

Thus  as  the  cross-ratio  of  any  flat  pencil  V[abed]  or  axial 
pencil  u{a/3yd)  may  be  taken  the  cross-ratio  of  the  cut  ABCD 
on  any  transversal. 

178.  Two  projective  primal  figures  are  'equicross;*  and 
inversely  two  equicross  primal  figures  are  projective. 

179.  As  D  approaches  the  point  at  infinity,  AD/BD  ap- 
proaches I.  The  cross-ratio  [ABCD]  when  D  is  figurative 
equals  AC/BC. 

180.  Given  three  costraight  points  ABC,  to  find  D  so  that 
[ABCD]  may  equal  a  given  number  ?t  (-{-  or  — ).  On  any 
straight  on  C  take  A'  and  B'  such  that  CA'/CB'  =n;  A'  and 
B'  lying  on  the  same  side  of  C  if  n  be  positive,  but  on  opposite 
sides  if  n  be  negative.  Join  AA\  BB\  crossing  in  V.  The 
parallel  to  A'B^  on  Fwill  cut  AB  in  the  required  D.  For  if 
D^  be  the  point  at  infinity  on  A'B\  and  ABCD  be  projected 
from  V,  then  A'B'CD'  is  a  cut  of  the  eject ;  so 

[ABCD]  =  [A'B'CU]  =A'C/B'C=.n, 

181.  If  [ABCD]  =  [ABCD;],  then  D,  coincides  with  D. 

182.  If  two  figures  be  complete  plane  perspectives,  four 
•costraight  points  (or  copunctal  straights)  in  one  are  equicross 
with  the  correlated  four  in  the  other.  Let  O  be  the  center  of 
perspective.  Let  M  and  M'  be  any  pair  of  correlated  points 
of  the  two  figures,  iVand  N'  another  pair  of  correlated  points 
lying  on  the  straight  OMM'  whose  cross  with  the  axis  of  per- 
spective is  X.     Then  [OXMN]  =  [6XM'N'\ 

That  is,    [OM/XM]/[ON/XN]  =  [OM' /XM']/[ON' /XN'\ 
Therefore  [OM/XM]/[OM' /XM']  =  [ON/XN]/[ON' /XN']. 
That  is,  [OXMM']  =  [OXNN'] ;  or  the  cross-ratio  [OXMM'] 


44  PROJECTIVE    GEOMETRY. 

is  constant  for  all  pairs  of  correlated  points  AT  and  M^  taken 
on  a  straight  OX  on  the  center  of  perspective. 

Next  let  L  and  L'  be  another  pair  of  correlated  points  and 
V  the  cross  of  OLL^  with  the  axis  of  perspective.  Since  LM 
and  VM^  cross  on  some  point  Z  of  the  axis  XV,  therefore  if 
OXMM  be  projected  from  Z,  the  cut  of  the  eject  by  OV  is 
O  YLL',  So  \pXMM'-\  =  [(9  YLL'^  ;  or  the  cross-ratio  l^XMM'^, 
is  constant  for  all  pairs  of  correlated  points. 

It  is  called  the  *  parameter'  of  the  correlation.  When  the 
parameter  equals  —  i,  the  range  OXMM'  is  harmonic,  and  two- 
correlated  elements  correspond  doubly,  are  coupled,  and  the 
correlation  is  'involutorial.' 

183.  When  the  correlation  is  involutorial  and  the  center  of 
perspective  is  the  figurative  point  on  a.  perpendicular  to  the 
axis  of  perspective,  this  is  called  the  *axis  of  symmetry,'  and 
the  complete  plane  perspectives  are  said  to  be  '  symmetrical.' 

184.  When  the  correlation  is  involutorial  and  the  axis  of 
perspective  is  figurative,  then  the  center  of  perspective  is  called 
the  'symcenter,'  and  the  complete  plane  perspectives  are  said 
to  be  '  symcentral.' 

Prob.  37.  In  a  plane  are  given  a  parallelogram  and  any  sect. 
With  the  ruler  alone  find  the  center  of  the  sect  and  draw  a  parallel 
to  it. 

Prob.  38.  The  locus  of  a  point  such  that  its  joins  to  four  given 
points  have  a  given  cross-ratio  is  a  conic  on  which  are  the  points. 

Prob.  39.  If  the  sides  of  a  trigram  are  tangent  to  a  conic,  the 
joins  of  two  of  its  fan-points  to  any  point  on  the  polar  of  the  third 
are  conjugate  with  respect  to  the  conic. 

Art  16.*    Homography  and  Reciprocation. 

185.  Two  planes  taken  as  both  point  fields  and  straight  fields, 
are  called  'homographic  '  (colHnear)  if  they  are  so  correlated  that 
to  each  point  on  the  one  (without  exception)  one  and  only  one 
point  on  the  other  corresponds,  and  vice  versa;  and  so  that  a 
point  and  straight  of  the  one  plane  which  belong  to  one  another 

*  This  article  follows  the  model  set  by  Enriques. 


HOMOGRAPHY  AND   RECIPROCATION.  45- 

correspond,  in  the  other  plane,  to  a  point  and  straight  belonging 
to  one  another.  The  relation  between  homographic  planes  is 
called  'homography'  (coUineation). 

An  homography  exists  between  a  plane  original  and  its  image 
[the  cut  of  its  eject  from  a  projection-vertex  not  on  it]. 

Honiography  is  the  most  general  transformation  which  trans- 
forms straights  into  straights. 

1 86.  Tw^o  planes,  taken  as  both  point  fields  and  straight 
fields,  are  called  'reciprocal'  when  they  are  so  correlated  that 
to  each  point  on  the  one  (without  exception)  one  and  only  one 
straight  on  the  other  corresponds  and  vice  versa;  and  so  that  to 
a  point  and  straight  of  the  one  plane  which  belong  to  one  another 
correspond  in  the  other  plane  a  point  and  straight  belonging  to 
one  another. 

To  costraight  points  in  the  first  plane  correspond  copunctal 
straights  in  the  second. 

187.  Two  sheaves,  taken  as  both  plane  sheaves  and  straight 
sheaves,  are  called  homographic  if  to  each  straight  of  the  one 
corresponds  a  straight  of  the  other,  and  to  each  plane  a  plane, 
and  vice  versa;  and  so  that  if  a  straight  and  plane  belong  to 
each  other  in  the  one,  so  do  their  correlatives  in  the  other. 

188.  Two  sheaves,  taken  as  both  plane  sheaves  and  straight 
sheaves,  are  reciprocal  if  to  each  straight  of  the  one  corresponds 
a  plane  of  the  other,  and  to  each  plane  a  straight,  and  vice  versa; 
and  so  that  if  a  straight  and  plane  belong  to  each  other  in  the 
one,  so  do  their  correlatives  in  the  other. 

189.  A  plane  and  sheaf  are  homographic  when  to  every 
point  of  the  plane  corresponds  a  straight  of  the  sheaf,  and  to 
every  straight  of  the  plane  a  plane  of  the  sheaf;  and  so  that  if 
the  point  and  straight  in  the  plane  belong  to  each  other,  so  do 
the  straight  and  plane  of  the  sheaf. 

190.  A  plane  and  sheaf  are  reciprocal  when  to  each  point 
of  the  plane  corresponds  a  plane  of  the  sheaf,  and  to  every 
straight  of  the  plane  corresponds  a  straight  of  the  sheaf;  and 
so  that  if  the  point  and  straight  belong  to  each  other  in  the  plane, 
the  plane  and  straight  of  the  sheaf  belong  to  each  other. 


46 


PROJECTIVE   GEOMETRY. 


Homography  and  reciprocation  are  included  together  as 
cases  of  projectivity. 

191.  In  two  homographic  planes  two  corresponding  ranges 
are  projective. 

192.  In  two  reciprocal  planes  a  range  is  projective  to  the 
corresponding  fiat  pencil. 

193.  Calling  the  plane  taken  as  a  point  field  and  a  straight 
field  and  the  sheaf  taken  as  a  plane  sheaf  and  a  straight  sheaf 
figures  of  the  second  class  or  secondary  figures,  then  between 
two  secondary  figures  there  is  a  fixed  projectivity  in  which  two 
pairs  of  projective  primal  figures  correspond  if  the  common 
elements  of  the  pairs  are  correlated. 

194.  Between  two  secondary  figures  there  is  a  projectivity 
fixed  by  four  pairs  of  corresponding  elements,  no  three  in  one 
primal  figure. 

195.  If  two  secondary  figures  are  homographic  one  can  be 
derived  from  the  other  by  a  finite  number  of  projectings  and 
cuttings. 

196.  To  depict  the  homography  between  two  planes  a  and  a' 
when  are  given  four  pairs 


of  corresponding  points,  dots  of 
two  tetrastims  A  BCD  and 
A'B'aD\ 

Set  up  a  projectivity  between  the 
connectors  of  corresponding  dots, 
for  example  between  AB  and  A^B', 
in  which  correspond  A  to  A\  B  to 
B'  and  the  codot  AB  .CD  to  the 
codot  A'B' .  CD'. 

Just  so  set  up  a  projectivity  be- 
tween the  flat  pencils  with  pencil- 
points  A  and  A'  in  which  to  the 
straights  AB,  AC,  AD  correspond 
A'B',  A'C,  A'Dr,  and  so  also  for 
those  with  pencil-points  B  and  B', 
etc. 


of  corresponding  straights,  sides  of 
two  tetragrams  abcddinda'h'dd'. 

Set  up  a  projectivity  between  the 
fans  of  corresponding  sides,  for 
example  between  ah  and  a'h'  in 
which  correspond  a  to  a',  h  to  ¥ 
and  the  diagonal  ah .  cd  to  the 
diagonal  a'b' .  c^d\ 

Just  so  set  up  a  projectivity  be- 
tween the  ranges  on  a  and  a'  in 
which  to  the  points  ab,  ac,  ad  cor- 
respond a'6',  a'c',  a^d',  and  so  also 
for  those  with  bearers  h  and  b%  etc. 


HOMOGRAPHY   AND   RECIPROCATION. 


47 


Now  in  a  let  any  straight  s  be 
given  not  on  one  of  the  points  Ay 
BfCjD;  then  it  will  cut  the  ranges 
AB  and  CD  in  two  points,  to  which 
the  corresponding  points  on  the 
ranges  ^'5'  and  C'Z)'  can  be  deter- 
mined; the  straight  5'  in  a'  which 
joins  these  points  will  be  the 
straight  5'  which  is  correlated  to 
the  straight  5  in  the  homography 
set  up  between  a  and  a\ 

In  a  if  on  the  other  hand  a  point 
P  be  given  not  in  one  of  the  con- 
nectors of  the  tetrastim  A  BCD, 
take  its  eject  from  A  and  its  eject 
from  B  and  find  the  straights  cor- 
responding to  these  in  the  pencils 
A^  and  B';  the  cross  of  these 
straights  will  be  the  point  P'  which 
is  correlated  to  the  point  P  in  this 
homography  set  up  between  a  and 
a'. 


Now  in  a  if  any  point  P  be  given 
not  on  one  of  the  straights  a,  b,  c, 
d,  take  its  eject  from  the  projec- 
tion vertex  ab  and  also  from  cdy 
and  then  determine  their  corre- 
sponding straights  in  the  flat  pen- 
cils a'6'  and  c'c?';  the  cross  P'  of 
these  will  be  the  point  P'  which  is 
correlated  to  P  in  the  homography 
set  up  between  a  and  a'. 

In  a  if  on  the  other  hand  a 
straight  s  be  given  not  in  a  fan  of 
the  tetragram  abed,  it  will  deter- 
mine a  flat  pencil  as  and  another, 
bs.  Take  those  in  a'  correspond- 
ing to  these.  Their  concur  will 
be  the  straight  s^  correlated  to  5 
in  the  homography  set  up  between 
a  and  a\ 


197.  To  depict  the  reciprocation  (    ^^^^    I  where  four  dots  of  a 

tetrastim  in  a  are  correlated  to  four  sides  of  a  tetragram  in  a', 
the  flat  pencils  Aj  B,  C,  D  are  made  projective  to  the  ranges 
Cf  b,  c,  dy  and  the  ranges  ABy  CD,  etc.,  to  the  flat  pencils  aby  cd,  etc. 

If  now  in  a  a  point  P  be  given  not  in  a  connector  oiABCDy 
then  take  its  eject  from  A  and  B  and  determine  the  points  corre- 
sponding to  these  on  a  and  6;  their  join  will  be  the  straight  in  a' 
correlated  to  P  in  the  reciprocation  between  a  and  a'. 

In  a  if  on  the  other  hand  a  straight  s  be  given  not  on  a  dot, 
determine  its  crosses  with  AB  and  CD  and  then  the  straights 
corresponding  to  these  in  the  flat  pencils  ab  and  cd)  their  cross 
will  be  the  point  P'  correlated  to  s  in  this  reciprocation  between 
a  and  a\ 


48  PROJECTIVE   GEOMETRY. 

198.  Perspective  Secondary  Figures. 

If  two  different  planes  are  perspective,  that  is,  correlated 
by  projection  from  an  outside  vertex,  then  their  meet  is  a  self- 
correlated  straight  and  bearer  of  a  range  of  self-correlated  points. 

199.  If  two  different  sheaves  are  perspective,  that  is  ejects 
of  the  same  plane,  then  the  planes  on  their  concur  are  self- 
correlated. 

200.  Inversely  we  have  the  theorem: 

If  two  different  planes  are  homo-  If  two  different  sheaves  are 
graphic  and  their  meet  the  bearer  homographic  and  their  common 
of  a  range  of  self-correlated  points,  axial  pencil  consists  wholly  of  self- 
the  planes  are  perspective.  correlated  planes,  the  sheaves  are 

perspective. 

Proof.  If  a  and  a'  are  the  planes  and  a  their  meet,  then 
every  straight  5  of  a  crosses  its  correlated  straight  s'  of  a'  in 
the  self- correlated  point  as. 

Now  let  A  and  B  be  two  points  of  a,  and  A'  and  W  the  two 
corresponding  points  of  a' .  The  straights  AB  and  A'W  are 
correlated  and  hence  cross  on  a;  thence  follows  that  A  A'  and  BB' 
are  coplanar  and  therefore  incident. 

Consequently  the  joins  of  corresponding  points  in  a  and  a' 
are  every  two  incident,  and  since  they  are  evidently  not  all  co- 
planar,  they  must  be  copunctal;   thus  a  and  a'  are  perspective. 

201.  Homology.  Consider  a  homography  between  two  co- 
planar  planes,  that  is  in  a  plane  a;  call  an  element  coinciding 
with  the  corresponding  element  a  double  element. 

If  four  points  of  the  plane  a,  no  three  costraight,  be  taken 
as  double  points,  a  homography  is  thus  set  up  called  the  identical 
homography;   in  it  every  element  is  self-correlated. 

So  in  a  non-identical  homography  of  the  plane  a  there  cannot 
be  four  double  points  no  three  costraight,  nor  four  double  straights 
no  three  copunctal. 

202.  In  a  the  join  af  two  double  points  is  from  the  homog- 
raphy a  double  straight  and  projectively  correlated  to  itself; 
if  then  there  be  on  it  a  third  double  point,  then  are  all  its  points 


HOMOGRAPHY  AND  RECIPROCATION.  49' 

double  points.     So  all  straights  of  a  flat  pencil  having  three 
double  straights  are  double  straights.     Hence  follows: 

If  in  a  plane  non-identical  homography  there  be  four  double 
points,  there  is  a  range  of  double  points ;  if  four  double  straights, 
there  is  a  flat  pencil  of  double  straights. 

203.  If  in  the  homography  there  be  a  range  n  of  double 
points,  its  bearer  n  crosses  every  straight  in  a  point  which,  as 
double  point,  must  belong  to  the  corresponding  straight,  that  is, 
any  two  corresponding  straights  cross  on  n. 

Inversely :  If  in  a  plane  homography  all  pairs  of  correspond- 
ing straights  cross  on  a  straight,  its  range  consists  of  double 
points,  since  every  point  of  it  is  pencil-point  of  a  double  flat 
pencil. 

204.  The  necessary  and  sufficient  condition  for  the  existence 
of  a  flat  pencil  of  double  straights  in  a  non-identical  plane  homog- 
raphy is  that  all  pairs  of  corresponding  points  are  costraight 
with  a  fixed  point. 

205.  Theorem.  Two  coplanar  homographic  planes  which 
have 

three  costraight  double  points,  and  three  copunctal  double  straights, 

hence  a  range,  w,  of  double  points,  and  hence  a  flat  pencil  of  double 

have  also  a  flat  pencil  of  double  straights,  have   also   a   range   of 

straights.  double  points. 

Proof.  All  pairs  of  corresponding  straights  a  and  a'  cross 
on  u.     In  fact  au  as  double  point  must  coincide  with  a'u. 

Put  through  u  a  plane  ai  different  from  a  (=«')  and  project 
a'  on  ai  from  an  outside  vertex  V.  There  results  an  homography 
between  ai  and  a,  for  which  w  is  a  range  of  self-correlated  points, 
hence  (§  200)  a  perspectivity;  consequently  the  pairs  of  corre- 
sponding points  MM\j  NN\  . . .  are  all  costraight  with  a  fixed 
point  U\,  Now  from  V  project  ai  back  upon  a'\  the  joins  of 
the  pairs  of  homologous  points  {MM\  NN^y .  .  .)  in  the  homog- 
raphy given  between  a  and  a'  will  now  all  be  copunctal  on  Uy 
the  image  oiU\. 


50  PROJECTIVE   GEOMETRY. 

So  U  is  the  pencil-point  of  a  flat  pencil  of  double  straights 
in  the  homography  between  a  and  a'. 

206.  The  special  plane  homography  (between  two  coplanar 
planes)  in  which  there  is  a  range  u  of  double  points  and  a  flat 
pencil  U  of  double  straights  is  called  Homology  with  the  axis  u 
and  the  center  U  (central  homography,  perspective  homography). 

In  it  corresponding 

straights    cross    on    the    axis    of     points  are  costraight  with  the  cen- 
homology.  ter  of  homology. 

207.  'Special  homology'  is  where  the  center  is  on  the  axis. 

208.  Particular  cases  of  homology  are  i)  afline  homology 
(perspective  affinity),  where  the  center  is  a  figurative  point  and 
the  axis  a  proper  straight;  2)  homothety  (perspective  similarity), 
where  the  axis  is  the  straight  at  infinity  and  the  center  a  proper 
point;   3)  translation,  where  both  axis  and  center  are  figurative. 

209.  Theorem.  There  is  a  plane  homology,  having  a  given 
axis  w,  and  a  given  center  Z7,  in  which  correspond 

two  points  A   and  A^  costraight  two  straights  a  and  a'  crossing  on 

with  U  (differing  from  it  and  not  u  (differing  from  it  and  not  on  the 

on  the  axis).  center). 

This  is  the  homography  deter-  This  is  the  homography  deter- 
mined by  the  assumption  that  u  mined  by  the  assumption  that  the 
is  self-correlated  and  on  it  exists  points  U  and  aa^  are  double  points 
the  identical  projectivity,  and  that  and  that  in  the  flat  pencil  U  (as 
the  straight  ^^'  is  self -correlated  set  up  by  the  homography)  exists 
and  on  it  (as  set  up  by  the  homog-  the  identical  projectivity^  and  in 
raphy)  exists  that  projectivity  in  the  flat  pencil  aa'  that  projectivity 
which  U  and  C=AA' .u  are  having  w  and  c=(ia' .  Z7  as  double 
double  points,  and  the  points  A  straights,  and  in  which  the  straights 
and  A^  correspond.  a  and  a'  correspond. 

The  point  B'  corresponding  to  a  The    straight   h'  corresponding 

given  point  B  outside  ^^'  is  the  to  a  given  straight  h  not  on  the 

cross    oi   BU   with    the   straight  point  aa^  is  the  join  of  the  point 

A^O  corresponding  to  AB,  where  a^o   corresponding    to    abj    where 

O^AB.u.  o=ab.U. 

210.  Involution.     In  a  plane  non-identical  homography  two 


HOMOGRAPHY    AND    RECIPROCATION.  51 

corresponding  elements  are  not  in  general  doubly  correlated, 
that  is,  if  to  the  element  A  corresponds  the  element  A',  then  to 
the  element  A^  corresponds  in  general  an  element  different  from 
A.  If  in  a  plane  homography  w  every  two  corresponding  elements 
are  doubly  correlated  (if  w=w-'^),  then  the  non-identical  homog- 
raphy is  called  involution. 

211.  If  in  a  homology  the  range  (A A'  UC)  [C=AA^  -u]  (and 
therefore  every  analogous  range)  is  assumed  to  be  harmonic,  the 
homology  (then  called  harmonic)  is  an  involution. 

Inversely,  considering  an  involution  in  a,  the  joins  of  two  cor- 
responding points,  as  A  and  A',  are  self-correlated,  and  so  there 
is  an  infinity  of  double  straights;  just  so  there  is  also  an  infinity 
of  double  points  as  crosses  of  pairs  of  corresponding  straights. 
But  if  in  a  non-identical  homography  there  are  more  than  three 
double  elements,  then  three  belong  to  a  primal  figure,  which  then 
consists  wholly  of  double  elements;  consequently  the  involution 
in  the  plane  a:  is  a  homology;  but  on  each  double  straight  not 
the  axis  the  corresponding  points  make  a  hyperbolic  involution; 
consequently  the  homology  is  harmonic. 

The  necessary  and  sufficient  condition  that  a  plane  homog- 
raphy should  be  an  involution  is  that  it  be  a  harmonic  homology. 

212.  Plane  Polarity.  In  general  in  a  reciprocation  between 
two  coplanar  planes  two  corresponding  elements  are  not  doubly 
correlated,  that  is,  to  a  point  A  corresponds  a  straight  a,  and 
this  a  in  the  given  reciprocation  corresponds  to  a  point  A'  dif- 
ferent from  A. 

A  plane  reciprocation  in  which  any  two  corresponding  elements 
are  doubly .  correlated,  that  is  a  reciprocation  identical  with  its 
inverse,  is  called  a  polarity;  a  point  and  straight  which  correspond 
in  a  plane  polarity  are  called  the  pole  and  polar  of  one  another^ 

Polarity  in  a  plane  may  also  be  defined  as  a  unique  reversible 
correlation  between  the  points  and  straights  such  that  if  the 
straight  (polar)  corresponding  to  A  is  on  Bj  the  polar  of  B  is 
on  A. 

213.  A  plane  reciprocation  is  a  polarity  if  there  be  a  triangle 
in  which   each   vertex   corresponds   to   th,e  opposite  side.     For 


52  PROJECTIVE    GEOMETRY. 

if  the  three  vertices  A,  B,  C  correspond  to  the  opposite  sides 
a,  b,  c,  then  must  AB  correspond  to  ab,  that  is  c  to  C,  etc.  Hence 
the  vertices  and  sides  are  doubly  correlated. 

Now  in  the  given  projectivity  the  range  a  is  projective  to 
the  flat  pencil  A  of  the  corresponding  straights,  so  that,  if  this 
pencil  be  cut  by  a,  we  obtain  on  it  a  projectivity;  since  in  this 
projectivity  the  points  B  and  C  are  doubly  correlated,  therefore 
it  is  an  involution ;  consequently  the  points  of  a  and  the  straights 
of  A  are  doubly  correlated. 

The  same  is  true  of  the  points  on  b  and  c  and  the  correspond- 
ing straights  on  B  and  C.  As  a  consequence  is  also  every  point 
P,  where  two  straights  a'  and  b^  on  A  and  B  cross,  doubly  corre- 
lated to  the  corresponding  straight  s,  fixed  as  the  join  of  A^  and 
JB'  (on  a  and  b)  which  correspond  to  a^  and  b\  Hence  the  recip- 
rocation considered  is  a  polarity. 

214.  In  a  plane  polarity,  triangles  whose  vertices  are  poles 
of  the  opposite  sides  are  called  self-conjugate  or  auto-polar  (or 
auto-reciprocal)  triangles.  There  is  an  infinity  of  auto-polar 
triangles  in  a  plane  polarity. 

215.  The  most  general  mode  of  obtaining  a  polarity  is  to 
designate  a  triangle  which  shall  be  auto-polar,  and,  not  on  a 
vertex,  a  straight  as  polar  of  a  point  not  on  a  side. 

216.  In  a  polarity  two  points  are  called  conjugate  or  reciprocal 
if  one  is  on  the  polar  of  the  other;  and  so  likewise  for  straights. 
A  point  on  its  own  polar  is  called  self-conjugate;  like  a  straight 
on  its  own  pole. 

217.  A  triangle  whose  three  vertices  or  three  sides  are  every 
two  conjugate  is  an  auto-polar  triangle  of  the  polarity. 

218.  If  a  point  ^  is  on  its  polar  a,  then  no  other  point  on 
a  is  self-conjugate. 

219.  No  straight  has  on  it  more  than  two  self -con  jugate  points. 
It  may  have  none. 

j220.  In  a  plane  polarity,  the  pairs  of  conjugate  points  on 
a   non-self-conjugate   straight    make   an   involution   either   con- 


HOMOGRAPHY   AND    RECIPROCATION.  53 

taining  no  self-conjugate  point  or  two  which  harmonically  separate 
the  pairs  of  conjugate  points. 

221.  In  a  plane  polarity,  if  ABC  be  an  auto-polar  triangle 
and  P  a  point  within  ABCy  then  if  p,  the  polar  of  P,  lies  wholly 
without  the  triangle  ABC,  the  polarity  has  no  self -con  jugate 
element  and  is  called  ''uniform"  since  in  it  every  involution 
of  conjugate  elements  is  elliptic. 

222.  In  a  plane  polarity,  if  ABC  be  an  auto-polar  triangle 
and  P  a  point  within  ABC,  then  if  p,  the  polar  of  P,  penetrate 
the  triangle  ABC,  the  polarity  has  self-conjugate  elements,  and 
is  called  non-uniform,  since  of  the  three  involutions  of  conjugate 
points  on  the  three  sides  of  an  auto-polar  triangle,  two  are  hyper- 
boUc  and  one  elliptic. 

223.  In  the  plane,  duahty  is  vaUd  for  all  visual  properties. 
In  the  sheaf,  duahty  is  vaUd  for  all  properties,  whether  visual 
or  metric. 

Prob.  40.  In  a  given  plane  polarity  consider  as  polars  all  the  tan- 
gents of  a  given  curve  C,  that  is  suppose  a  polar  to  envelop  the  given 
curve ;  then  its  pole  will  define  another  curve  C  whose  points  are  the 
poles  of  the  tangents  of  C.  Reciprocally  the  points  of  C  are  the  poles 
of  the  tangents  of  C\ 

224.  Two  curves  C  and  C  such  that  each  is  the  locus  of  the 
poles  of  the  tangents  of  the  other,  and  Hkewise  the  envelope  of 
the  polars  of  the  points  of  the  other,  are  called  polar  reciprocals 
one  of  the  other  with  respect  to  the  polarity  P. 

225.  The  degree  or  order  of  a  curve  is  given  by  the  greatest 
number  of  points  in  which  it  can  be  cut  by  any  arbitrary  plane 
(for  a  plane  curve,  by  any  coplanar  straight). 

226.  The  class  of  a  plane  curve  is  given  by  the  greatest  number 
of  tangents  which  can  be  drawn  to  it  from  any  arbitrary  point 
in  the  plane. 

Prob.  41.  The  degree  and  class  of  a  curve  are  equal  to  the  class 
and  degree  respectively  of  its  polar  reciprocal. 

Prob.  42.  The  polar  reciprocal  of  a  conic  is  a  conic. 

227.  Two  reciprocal  figures  are  duals  which  have  a  definite 


54  PROJECTIVE    GEOMETRY. 

special  relation  to  one  another  with  respect  to  their  positions, 
while  on  the  other  hand  if  two  figures  are  merely  duals,  there 
is  no  relation  of  any  kind  between  them  as  regards  their  position. 

Prob.  43.  If  two  triangles  are  both  auto-polar  with  respect  to  a 
given  conic,  their  six  vertices  are  on  a  conic,  and  their  six  sides  touch 
another  conic. 

Prob.  44.  If  a  conic  C  touch  the  sides  of  a  triangle  abc  auto-polar 
with  regard  to  another  conic  K,  there  is  an  infinity  of  other  triangles 
auto-polar  with  regard  to  K  which  circumscribe  C 

Prob.  45.  Two  triangles  circumscribing  the  same  conic  have  their 
vertices  on  another  conic.  If  two  triangles  are  inscribed  in  the  same 
conic,  their  six  sides  touch  another  conic. 

Prob.  46.  If  a  conic  C  circumscribe  a  triangle  auto-polar  with  re- 
spect to  another  conic  K^  there  is  an  infinity  of  other  triangles  inscribed 
in  C  and  auto-polar  with  respect  to  K\  and  the  straights  which  cut  C 
and  K  in  harmonic  conjugates  touch  a  third  conic  C,  the  polar  recip- 
rocal of  C  with  regard  to  K. 

Prob.  47.  If  a  triangle  inscribed  in  one  conic  circumscribes  another 
conic,  then  there  is  an  infinity  of  such  triangles. 

Prob.  48.  Two  triangles  reciprocal  with  respect  to  a  conic  are  in 
homology. 

Prob.  49.  Two  triangles  in  homology  determine  a  polarity,  in  which 
the  center  of  homology  is  the  pole  of  the  axis  of  homology  and  any 
vertex  is  the  pole  of  the  corresponding  side  of  the  other  triangle.  If 
no  point  is  self-conjugate,  this  is  not  a  polarity  with  respect  to  a  conic. 

If  a  point  be  self-conjugate,  construct  the  conic  with  regard  to  which 
the  triangles  are  reciprocal. 


Art.  17.    Transformation.    Pencils  and  Ranges  of  Conics. 

228.  If  any  two  points  be  assumed  to  determine  not  only  a 
straight  but  also  a  sect,  that  point -row  on  their  straight  of  which 
they  are  the  end  points,  then  the  sect  determined  by  a  fixed  point 
O,  the  origin,  and  a  point  Z,  may  be  represented  by  x. 

Then  if  two  conjective  ranges  be  correlated,  this  correlation 
is  defined  by  the  equation 


TRANSFORMATION.      PENCILS   AND    RANGES   OF   CONICS.  55 

where  x'  is  the  sect  OX',  X'  corresponding  to  X.  Hence  x= 
—  {cx'-\-d)l{ax'-\-h)\  therefore  if  K,  Z,  My  N  be  four  costraight 
points  and  K^,  U,  M\  N',  their  conjective  correlates, 

Proof.     PQ  =  :r2  -  xi  =  (X2^ — Xi^)  (ad  —  hc)/{axi'  +  h)  {ax  J  +  h) ; 
SR  =  x^  —  X4,=  (X3' — x^)  {ad  —  bc)/{ax2f  +  b)  {ax^!  +  &) , . 
etc.;  hence 

[KLMN]  =  {xi  -  X2)  {X4  -  X3)/{X2  -  X3)  {xi  -  X4) 

=  {xi'  -  xd)  {xi  -  x^)l{x^  -  xj)  {x^  -  xi)  =  \k'L'M'lSl'\ 

So  if  three  pairs  of  points  be  mated  the  correlation  is  deter- 
mined. 

229.  United  Points.  Making  :v  =  ^'  we  have  ax'^-\-(b-\-c)x^-d 
=  0;  hence  in  every  conjective  correlation  the  coincidence  of  a 
point  with  its  corresponding  point  will  occur  twice,  that  is  there 
are.  two  united  points,  27,  t/'  (real  or  imaginary). 

If  the  origin  O  be  the  center  of  UU'  =  Uj  then  6  +  c  =  o;  hence 

(i)     axx'-\-h{x  —  x')-\-d  =  Oj    and     (2)     a{ul2f-\rd=o. 
Combining  (i)  and  (2),  the  equation  of  correlation  becomes 

xo^  -\-{x— oc')h/a  —  (w/2)2  =  o, 
.-.  {x-\-u!2){xf -u/2)  =  {x' -x){u/2+h/a), 
.-.  u/{u/2+h/a)  =  {x-x'){-u)/{x'-u/2){x-\-u/2) 

=  XX'  •  UU'/{X'U-XU')  =  [XX'UU'l 

So  the  cross-ratio  of  a  point,  its  corresponding  point,  and 
the  united  points  is  constant.  The  correlation  is  therefore  deter- 
mined if  its  united  points  and  one  pair  of  corresponding  points 
be  given. 

230.  Double  Points  and  Involution.  If  b  =  Cj  then  axxf  •\- 
h{x-\-x!)-\-d=o\  hence  in  whichever  of  the  two  ranges  a  point 
be  taken,  it  has  the  same  mate;  hence  the  elements  are  coupled, 
the  correlation  is  involutoric. 

The  equation  may  be  written  a{x  +  hid)  {xf  +  h/d)  =  (1^  —  ad) fay 
which  gives  for  the  united  points  the  values  —h  a±{h'^  —  ad)^/a. 


56  PROJECTIVE   GEOMETRY. 

So  if  M  be  the  center  of  UU',  then  0M=  -b/a. 
MX-MX'  =  MU^;    hence  the  'double'  points  U  and  U'oi 
the  involution  separate  harmonically  any  couple. 

231.  An  operation  which  replaces  a  given  figure  by  a  second 
figure  in  accordance  with  a  given  law  is  called  a  'transformation.' 

If  a  transformation  replaces  the  points  of  one  figure  by  the 
points  of  a  second,  it  is  called  a  'point  transformation.' 

If  a  point  transformation  replaces  X(x,  y)  by  Z'(:x/,  /),  then 
the  equations  expressing  o(/  and  y'  in  terms  of  x  and  y,  or  inversely, 
are  called  the  'equations  of  the  transformation.' 

If  the  corresponding  costraight  points  have  the  same  cross 
ratio,  the  transformation  is  called  'projective.'  We  have  seen 
that  x^  =  (fnx+n)/(mix+ni)  is  the  equation  of  a  projective  trans- 
formation. 

For  homography,  the  general  projective  transformation  of 
the  plane,  the  equations  are 

x^  =  (aiX  +  biy+ci)/(a3X  +  b3y+C3)j        '      . 
y  =  {a2X  +  b2y  +  C2)/{a3X-{-bsy+C3). 

232.  The  assemblage  of  conies  on  which  are  the  dots  A,  By 
C,  D  of  a  given  tetrastim  is  called  the  'pencil  of  conies'  through 
the  'basal  points'  A,  B,  C,  D.  The  three  pairs  of  opposite 
connectors  of  the  tetrastim  are  called  the  'degenerate  conies'  of 
the  pencil,  and  determine  on  any  transversal  an  involution  in 
which  its  intersection-points  with  any  conic  of  the  pencil  are 
a  couple.     (Desargues-Sturm  theorem.) 

233.  The  assemblage  of  conies  on  which  are  the  sides  a,  b,  c,  d 
of  a  given  tetragram  is  called  the  'range  of  conies'  touching  the 
^ basal  straights'  a,  b,  c,  d.  The  three  pairs  of  opposite  fan- points 
of  the  tetragram  are  called  the  degenerate  conies  of  the  range, 
and  determine  on  any  external  point  not  on  a  side  of  the  tetragram 
an  involution  in  which  its  tangents  to  any  conic  of  the  range 
are  a  couple. 

234.  All  polars  of  a  point  P  with  respect  to  the  conies  of  a 
pencil  are  copunctal  [in  Q,  and,  inversely,  of  Q,  in  P].  (Both 
•points  are  called  conjugate  with  regard  to  the  conies  of  the  pencil.) 


($^0-^   ^1 


<l^fYJ^^ 


J 

TRANSFORMATION.      PENCILS   AND   RANGES   OF   CONICS.  57 

Prob.  50.  Pole-conic  of  a  straight.  If  a  point  P  describe  a  straight 
5,  then  the  intersection  point  P'  of  its  polars  with  respect  to  the  conies 
of  a  pencil  describes  a  conic.  (This  is  also  the  locus  oif  the  poles  of 
s  with  regard  to  the  individual  conies  of  the  pencil.) 

Prob.  51.  Newton.  There  are  two  conies  which  go  through  four 
given  points  and  touch  a  given  straight. 

Prob.  52.  Through  a  given  point  there  are  two  conies  tangent  to 
the  sides  of  a  given  tetragram,  or  none. 

Prob.  53.  To  determine  whether  there  is  a  conic  through  the  dots 
of  a  given  tetrastim  A  BCD  and  tangent  to  a  given  straight  s,  it  suffices 
to  determine  whether  the  point-pairs  in  which  s  is  cut  by  two  pairs  of 
opposite  connectors  of  A  BCD  separate  each  other. 

Prob.  54.  Midpoints  conic.  The  centers  of  the  conies  of  a  pencil 
are  on  a  conic,  whose  center  is  the  mass-center  of  the  basal  points. 

Prob.  55.  The  codots  of  a  tetrastim  and  the  centers  of  its  six  con- 
nectors are  on  one  conic. 

235.  All  poles  of  a  straight  with  respect  to  the  conies  of  a 
range  are  costraight. 

Prob.  56.  If  a  straight  s  rotates  about  a  fixed  point,  then  the  bearer 
of  all  its  poles  with  respect  to  the  conies  of  a  range  envelops  a  conic. 

236.  Midpoints  straight.  (Newton.)  The  centers  of  the 
conies  of  a  range  are  eostraight,  on  the  join  of  the  centers  of  the 
diagonals  of  the  basal  tetragram. 

Prob.  57.  The  centers  of  the  three  diagonals  of  a  tetragram  are  co- 
straight. 

237.  The  intersections  of  a  pencil  of  conies  and  a  projective 
flat  pencil  are  a  curve  of  the  third  order  containing  the  pencil- 
point  and  the  basal  points. 

238.  Every  cubic  is  the  intersection  of  a  pencil  of  conies  and  a 
projective  flat  pencil. 

239.  A  range  of  conies  combined  with  a  projective  point-row 
gives  a  curve  of  the  third  class  touching  the  bearer  of  the  point- 
row  and  the  basal  straights. 

240.  Every  curve  of  the  third  class  is  the  envelope  of  the 
system  of  straights  obtained  from  combining  a  range  of  conies 
v^ith  a  projective  po'nt-row. 


58  PROJECTIVE   GEOMETRY. 

241.  Definition: 

Four  conies  of  a  pencil  are  called  Four  conies  of  a  range  are  called 

'harmonic'   if  the  polars  of  any  'harmonic'    if   the   poles   of   any 

point  with  respect  to  them  are  four  straight  with  respect  to  them  are 

harmonic  straights.  four  harmonic  points. 

242.  Two  projective  pencils  of  conies  produce  a  curve  of  the 
fourth  order. 

243.  Every  curve  of  the  fourth  order  may  be  produced  by 
two  projective  pencils  of  conies. 

Prob.  58.  Two  projective  ranges  of  conies  produce  a  curve  of  the 
fourth  class. 

244.  Every  curve  of  the  fourth  class  may  be  produced  by 
two  projective  ranges  of  conies. 

Prob.  59.  The  codot  tristim  of  the  basal  tetrastim  of  a  pencil  of 
conies  is  auto-polar  with  regard  to  each  of  them.  So  is  the  diagonal 
trigram  of  the  basal  tetragram  of  a  range  of  conies. 

Prob.  60.  The  conies  of  a  pencil  determine  on  two  straights  through 
one  basal  point,  or  through  two  basal  points,  projective  point-rows. 

Prob.  61.  An  arbitrary  straight  of  its  plane  is  touched  by  only  one 
conic  of  a  range,  but  two  go  through  an  arbitrary  point. 


INDEX, 


Affine,  50. 
Angle,  9. 
Apex,  9. 
Asymptote,  33. 

cone,  40. 
planes,  37. 
Auto-polar,  52. 

-reciprocal,  52. 
Axial,  9. 

angle,  9. 
eject,  9. 
parallel,  11. 
pencil,  9. 
projection,  9. 
-Axis,  9,  15. 

of  perspective,  15. 
projection,  9. 

Basal,  56. 
Bearer,  6. 
Bolyai,  42. 
Braikenridge,  25. 
Brianchon,  22. 

point,  22. 

■Center,  32. 

of  a  conic,  32. 

of  an  involution,  27. 

of  perspective,  15. 

of  sect,  32. 
Central  homography,  50. 
Chasles,  24. 
Chord,  32. 
Class,  24,  40. 


Clebsch,  24. 
Coaxal,  14. 
Codot,  13. 

tristim,  14. 
CoUinear,  44. 
Collineation,  45. 
Compendent,  6,  11. 
Complete  plane  perspective,  14. 

set,  13. 
Cone,  20. 

of  planes,  20. 
Conic,  construction  of,  21. 

curve,  24. 

degenerate,  56. 

pencil,  20. 

range,  20. 

surface,  20. 
Conies,  pencil  of,  56. 
range  of,  56. 
Conjective,  27. 
Conjugate  diameters,  32. 

points,  17,  25,  52,  56. 
straights,  25. 
Conjugates,  harmonic,  17. 
Constant,  55. 
Contact,  22. 
Copolar,  14. 
Copunctal,  9. 
Correlation,  12. 

involutoric,  55. 
Costraight,  9. 
Coupled,  27. 
Cross,  6. 
Cross-ratio,  42. 


60 


INDEX. 


Cubic,  40. 
Culmann,  12. 
Curve,  53. 

class  of,  53. 

cubic,  40. 

degree  of,  53. 

of  fourth  order,  58. 

of  second  order,  20. 

of  third  class  57. 

order  of,  53. 

plane,  53. 

twisted,  40. 
Cut,  10. 
Cyclic,  6. 

Darboux,  4. 

Degenerate,  56. 

Degree,  53. 

Depict,  47- 

Desargues  theorem,  28. 

Desargues-Sturm  theorem,  56. 

Descriptive  geometry,  43.      ^ 

Diagonal,  13. 

trigram,  14. 
Diameter,  32. 
Dots,  12. 
Double  elements,  27. 

points,  51. 
Dual  12. 
Duality,  12,  34. 
Eject,  10. 
Elements,  8. 
Ellipse,  24. 
Elliptic,  28. 
Equation  of  correlation,  54. 

of  transformation,  56. 
Equicross,  43. 
Euchd,  4. 
Euclidean  geometry,  42. 

Fans,  12. 
Figurative,  6,  11. 
Flat  pencil,  9. 
Fundamental  theorem,  15. 

Geometry  descriptive  43 


I  Geometry,  Euclidean,  42. 

Non-Euclidean,  42, 

points,  41. 

straights,  41. 
Graphic  statics,  12. 
Guide-straight,  39. 

Harmonic  axial  pencil,  16. 

conies,  58. 

conjugates,  17. 

elements,  15. 

flat  pencil,  16. 

homology,  51. 

planes,  16. 

points,  15,  29. 

range,  15. 

straights,  16. 
Hexagram,  15,  22. 

skew,  41. 
Hexastim,  15,  22. 
nomographic,  44. 
Homography,  44. 
Homology,  48. 

special,  50. 
Homothety,  50. 
Hyperbola,  24. 
Hyperbolic,  28. 

paraboloid,  37. 
Hyperboloid  of  one  nappe,  38. 

Image,  10. 

Imaginary  points,  55. 
Infinity,  10. 
Intersection,  9. 
Involution,  26,  50. 
Involution  axis,  31. 
Involution  center,  31. 
Involutorial  or  involutoric,  55. 

Join,  8. 
Junction,  8. 

Kern  curve,  26. 

Lindemann,  42. 
Lobachevski,  42. 
Locus,  42,  44. 


INDEX. 


61 


Maclaurin,  23. 

configuration,  23. 
Mates,  12. 
Meet,  9. 
Metric,  50. 
Midpoints  conic,  57. 
Midpoints  straight,  57. 
Mole,  4. 
Monge,  41. 

Newton,  25,  57. 
Non-Euclidean  geometry,  42. 

Opposite,  14. 
Order,  24. 
Original,  10. 

Pappus,  42. 
Parabola,  24. 
Parallel,  11. 
Parallel  axial,  ir. 

flat  pencil,  11. 
Parameter,  44. 
Pascal,  22. 

straight,  22. 
Pass,  9. 
Pencil,  9. 

axial,  9. 

conic,  20. 

flat,  9. 

of  conies,  56. 

of  second  class,  20. 

point,  9. 
Pentagram,  23. 
Pentastim,  23. 
Permutation,  6. 
Perspective,  19,  29,  48. 
Picture-plane,  10. 
Plane,  8. 

-sheaf,  34. 
-space,  34 
Poncelet,  42. 
Point,  8. 

Brianchon,  22. 
of  contact,  22. 

-field,  34. 
figurative,  11. 


Point,  proper,  11. 
-row,  34. 
-space,  34. 
Points,  8. 

basal,  56. 

geometry,  41. 

harmonic,  15. 
Polar,  25. 
Polar  curve,  26. 
Polarity,  51.  ' 

Pole,  15,  25. 

-conic,  57. 
Polygram,  12. 
Polystim,  12. 
Primal,  9. 

Principle  of  duality,  12,  35. 
Project,  10. 
Projected,  10. 
Projection-axis,  10. 
Projection-vertex,  10. 
Projecting  planes,  10. 

straights,  10. 
Projective  conic  ranges,  29. 
Projectivity,  18. 
Projector,  10. 
Proper,  11. 

Quadric  surface,  $6. 
Quadrilateral  construction,  4,  16.. 

Range,  9. 

of  conies,  56. 

of  second  degree,  20. 
Rays,  II. 
Reciprocal,  45. 
Reciprocation,  44. 
Regulus,  36. 
Reye,  12. 
Riemann,  42. 
Ruled,  doubly,  40. 

surface,  s^, 

system,  36. 

Secondary  figures,  46. 
Sect,  9. 

unit,  42 


62 


INDEX. 


Self-conjugate,  52. 
Sense,  11,  42. 
Separate,  6. 
Sequence,  7 
Sheaf,  45. 

plane,  34. 

straight,  34. 
Sides,  9,  12. 
Skew,  41, 
Space,  four-dimensional,  41. 

of  straights,  41. 

plane-,  34. 

point-,  34. 
Steiner,  24. 
Straight,  8. 

basal,  56. 
-field,  35. 
-sheaf,  35. 
Straights-geometry,  41. 
Subject,  10. 
Surface,  quadric,  ^6. 
ruled,  36. 

of  second  degree,  38. 
Symcenter,  44. 
Symcentral,  44. 
Symmetrical,  44. 
Symmetry,  axis  of,  44. 


System,  ruled,  36. 

Tactile  space,  4. 
Tangent,  22,  39. 

plane,  39. 
Tetragram,  14. 
Tetrastim,  14.  , 

Throw,  26. 
Transfix,  10. 
Transformation,  56. 

point,  56. 

projective,  56. 
Translation,  50. 
Transversal,  10. 
Trigram,  14. 
Tristim,  14. 
Twisted  cubic,  40. 

Uniform,  53. 
United  points,  55. 

Vanishing-point,  27, 
Visual  properties,  53. 

space,  4. 
von  Staudt,  4,  42. 

Within,  24. 
Without,  24- 


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4 


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VoL*I Large  8vo, 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo, 

O'DriscoU's  Notes  on  the  Treatment  of  Gold  Ores 8vo, 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) i2mo, 

"  "  "  "  Part  Two.     (TurnbulL) i2mo, 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo,  paper, 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) Svo, 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) 12 mo, 

Poole's  Calorific  Power  of  Fuels Sv.o, 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,    i  25 

*  Reisig's  Guide  to  Piece-dyeing Svo,  25  00 

Richards  and  Woodman's   Air,  Water,  and   Food  from  a  Sanitary  Stand- 
point  Svo, 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo. 

Cost  of  Food,  a  Study  in  Dietaries i2mo, 

*  Richards  and  Williams's  The  Dietary  Computer Svo, 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. 

Non-metalUc  Elements.) Svo,  morocco, 

Ricketts  and  Miller's  Notes  on  Assaying Svo, 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage Svo, 

Disinfection  and  the  Preservation  of  Food Svo, 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory. Svo, 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) Svo, 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo, 

Ruddiman's  Incompatibilities  in  Prescriptions Svo, 

*  Whys  in  Pharmacy i2mo, 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo, 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Omdorff.) Svo, 

Schimpf's  Text-book  of  Volumetric  Analysis i2mo. 

Essentials  of  Volumetric  Analysis i2mo, 

*  Qualitative  Chemical  Analysis Svo, 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco, 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco, 

Stockbridge's  Rocks  and  Soils Svo, 

*  TiUman's  Elementary  Lessons  in  Heat Svo, 

*  Descriptive  General  Chemistry Svo, 

Treadwell's  Qualitative  Analysis.     (HalL) Svo, 

Quantitative  Analysis.     (Hall.) Svo, 

Turneaure  and  Russell's  Public  Water-supplies Svo, 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo, 

*  Walke's  Lectures  on  Explosives Svo, 

Ware's  Beet-sugar  Manufacture  and  Refining Small  Svo,  cloth, 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks Svo, 

Wassermann's  Immtme  Sera :  Hemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)  i2mo. 

Well's  Laboratory  Guide  in  QuaUtative  Chemical  Analysis Svo, 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students i2mo. 

Text-book  of  Chemical  Arithmetic i2mo, 

Whipple's  Microscopy  of  Drinking-water Svo, 

Wilson's  Cyanide  Processes i2mo, 

Chlorination  Process i2mo, 

Winton's  Microscopy  of  Vegetable  Foods Svo, 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry i2mo, 

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CIVIL  ENGINEERING. 

BRIDGES    AND   ROOFS.       HYDRAULICS.       MATERIALS   OF   ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments lamo, 

Bixby's  Graphical  Computing  Table Paper  19^X24!  inches. 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) 8vo, 

Comstock's  Field  Astronomy  for  Engineers 8vo, 

Davis's  Elevation  and  Stadia  Tables 8vo, 

Elliott's  Engineering  for  Land  Drainage i2mo. 

Practical  Farm  Drainage i2mo, 

♦Fiebeger's  Treatise  on  Civil  Engineering , 8vo, 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo, 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo, 

French  and  I/es's  Stereotomy 8vo, 

Goodhue's  Municipal  Improvements i2mo, 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo, 

Gore's  Elements  of  Geodesy Svo, 

Haj^ord's  Text-book  of  Geodetic  Astronomy Svo, 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco, 

Howe's  Retaining  Walls  for  Earth i2mo, 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  Svo, 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods '.  . .  .8vo, 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscoit  and  Emory.) .  i2mo, 
Mahan's  Treatise  on  Civil  Engineering,     (1873.)     (Wood.). .8vo, 

*  Descriptive  Geometry Svo, 

Merrlman's  Elements  of  Precise  Surveying  and  Geodesy Svo, 

Merriman  and  Brooks's  Handbook  for  Surveyors l6mo,  morov,^. 

Nugent's  Plane  Surveying Svo, 

Ogden's  Sewer  Design i2mo, 

Patton's  Treatise  on  Civil  Engineering Svo  half  leather, 

Reed's  Topographical  Drawing  and  Sketching 4to, 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage Svo, 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry Svo, 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Svo, 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

Svo, 
Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced Svo, 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco. 

Wait's  Engineering  and  Architectural  Jxurisprudence Svo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo, 

Sheep, 

Law  of  Contracts 8vo, 

Warren's  Stereotomy — Problems  in  Stone-cutting Svo, 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco, 

Wilson's  Topographic  Surveying 8vo, 


BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges . .  Svo,    2  00 

*       Thames  River  Bridge 4to,  paper,    5  00 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,    3  5o 

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Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo,  3  00 

Design  and  Construction  of  Metallic  Bridges 8vo,  5  00 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  00 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  00 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches 8vo,  4  00 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel.  . 8vo,  2  00 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  00 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges : 

Part  I.     Stresses  in  Simple  Trusses 8vo,  2  50 

Part  II.     Graphic  Statics 8vo,  2  50 

Part  III.     Bridge  Design 8vo,  2  50 

Part  IV.     Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge 4to,  10  00 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,  2  00 

Specifications  for  Steel  Bridges i2mo,  i  2S 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume 8vo,  3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo, 

Bovey's  Treatise  on  Hydraulics 8vo, 

Church's  Mechanics  of  Engineering 8vo, 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper. 

Hydraulic  Motors 8vo, 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco, 

Flather's  Dynamometers,  and  the  Meastirement  of  Power i2mo, 

Folwell's  Water-supply  Engineering.  . 8vo, 

Frizell's  Water-power 8vo, 

Fuertes's  Water  and  Public  Health i2mo. 

Water-filtration  Works i2mo, 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo, 

Hazen's  Filtration  of  Public  Water-supply 8vo, 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo, 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo, 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo, 

Merriman's  Treatise  on  Hydraulics 8vo, 

*  Michie's  Elements  of  Analytical  Mechanics 8vo, 

Schuyler's  Reservoirs  for  Irrigation,  Water-power,  and  Domestic  Water- 
supply Large  8vo, 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post.,  44c.  additional. ).4to, 

Turneaure  and  Russell's  Public  Water-supplies 8vo, 

Wegmann's  Design  and  Construction  of  Dams 4to, 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4to, 

Williams  and  Hazen's  Hydraulic  Tables 8vo, 

Wilson's  Irrigation  Engineering Small  8vo, 

Wolff's  Windmill  as  a  Prime  Mover 8vo, 

Wood's  Turbines 8vo, 

Elements  of  Analytical  Mechanics 8vo, 

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MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo, 

Roads  and  Pavements 8vo, 

Black's  United  States  Public  Works .Oblong  4to, 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo, 

Btirr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo, 

Byrne's  Highway  Construction 8vo, 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo, 

Church's  Mechanics  of  Engineering 8vo, 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to, 

♦Eckel's  Cements,  Limes,  and  Plasters 8vo, 

Johnson's  Materials  of  Construction Large  8vo, 

Fowler's  Ordinary  Foundations 8vo, 

*  Greene's  Structural  Mechanics 8vo, 

Keep's  Cast  Iron 8vo, 

Lanza's  Applied  Mechanics 8vo, 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo, 

Maurer's  Technical  Mechanics 8vo, 

Merrill's  Stones  for  Building  and  Decoration 8vo, 

Merriman's  Mechanics  of  Materials 8vo, 

Strength  of  Materials i2mo, 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo, 

Patton's  Practical  Treatise  on  Foundations 8vo, 

Richardson's  Modern  Asphalt  Pavements 8vo, 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor., 

Rockwell's  Roads  and  Pavements  in  France i2mo, 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo, 

Smith's  Materials  of  Machines i2mo. 

Snow's  Principal  Species  of  Wood 8vo, 

Spalding's  Hydraulic  Cement i2mo, 

Text-book  on  Roads  and  Pavements i2mo, 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo, 

Thurston's  Materials  of  Engineering.     3  Parts 8vo, 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo, 

Part  II.     Iron  and  Steel 8vo, 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo, 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo, 

Tillson's  Street  Pavements  and  Paving  Materials 8vo, 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.) .  .  i6mo,  mor.. 

Specifications  for  Steel  Bridges i2mo. 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo, 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo, 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,    4  00 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Brook's  Handbook  of  Street  Raihoad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book. .  i6mo,  morocco,  5  00 

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Dredge's  History  of  the  Pennsylvania  Railroad:   (1879) Paper,  5  00 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  00 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explores*  Guide. .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  I  00 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  00 

Nagle's  Field  Manual  for  Raikoad  Engineers i6mo,  morocco,  3  00 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  00 

Searles's  Field  Engineering i6mo,  morocco,  3  00 

Raikoad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork Svo,  i  50 

*  Trautwine's  Method  of  Calculating  the.  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams Svo,  2  00 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Raikoads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Raikoad  Censtruction i6mo,  morocco,  5  00 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  Svo,  s  00 


DRAWING. 

Barr's  Kinematics  of  Machinery Svo, 

*  Bartlett's  Mechanical  Drawing Svo, 

*  "  "  "        Abridged  Ed Svo, 

Coolidge's  Manual  of  Drawing Svo,  paper 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to, 

Durley's  Kinematics  of  Machines Svo, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications Svo, 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective Svo, 

Jamison's  Elements  of  Mechanical  Drawing Svo, 

Advanced  Mechanical  Drawing Svo, 

Jones's  Machine  Design : 

Part  I.     Kinematics  of  Machinery Svo, 

Part  n.     Form,  Strength,  and  Proportions  of  Parts Svo, 

MacCord's  Elements  of  Descriptive  Geometry Svo, 

Kinematics;  or.  Practical  Mechanism Svo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams Svo, 

MacLeod's  Descriptive  Geometry Small  Svo, 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting Svo, 

Industrial  Drawing.     (Thompson.) Svo, 

Moyer's  Descriptive  Geometry 8vo, 

Reed's  Topographical  Drawing  and  Sketching 4to, 

Reid's  Course  in  Mechanical  Drawing 8vo, 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo, 

Robinson's  Principles  of  Mechanism 8vo, 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo, 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.     (McMillan.) Svo, 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo, 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo. 

Drafting  Instruments  and  Operations i2mo. 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo. 

Plane  Problems  in  Elementary  Geometry i2mo, 

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Warren's  Primary  Geometry i2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50. 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo- 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry Svo,  2  50 

Weisbach's    Kinematics  :and    Power    of    Transmission.        (Hermann    and 

Klein.) Svo,  5  Oq, 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo,  2  00 

Wilson's  (H.  M.)  Topographic  Surveying Svo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective Svo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering Svo,  i  oo- 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  Svo,  3  oa 


ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  Svo, 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  . .  .  i2mo, 
Benjamin's  History  of  Electricity Svo, 

Voltaic  Cell Svo, 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo, 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo, 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco, 
Dolezalek's    Theory   of    the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende. ) i2mo, 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) Svo, 

Flather's  Dynamometers,  and  the  Measurement  of  Power .i2mo, 

Gilbert's  De  Magnete.     (Mottelay.) Svo, 

Hanchett's  Alternating  Currents  Explained i2mo, 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco, 

Holman's  Precision  of  Measurements Svo, 

Telescopic   Mirror-scale  Method,  Adjustments,  and   Tests.  . .  .Large  Svo, 

Kinzbrunner's  Testing  of  Continuous-current  Machines Svo, 

Landauer's  Spectrum  Analysis.     (Tingle.) Svo, 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo. 
Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) Svo, 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  H.  Svo,  each, 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light Svo, 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo, 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .Svo, 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  I Svo, 

Thurston's  Stationary  Steam-engines. Svo, 

*  Tillman's  Elementary  Lessons  in  Heat Svo, 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  Svo, 

Ulke'-s  Modern  Electrolytic  Copper  Refining Svo, 


LAW. 

*  Davis's  Elements  of  Law Svo, 

*  Treatise  on  the  MiUtary  Law  of  United  States Svo, 

*  Sheep, 

Manual  for  Courts-martial i6mo,  morocco. 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo 

Sheep, 

Law  of  Contrasts 8vo, 

Winthrop's  Abridgment  of  Military  Law i2mo, 

10 


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2  SO 

MANUFACTURES. 

Bernadou's  Smokeless  Powder— Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Bolland's  Iron  Founder i2mo,  2  50 

"The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

Efifront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  00 

Fitzgerald's  Boston  Machinist i2mo,  i  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Hopkin's  Oil-chemists*  Handbook Svo,  3  00 

Keep's  Cast  Iron Svo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  Svo,  7  50 

Matthews's  The  Textile  Fibres Svo,  3  50 

Metcalf's  SteeL     A  Manual  for  Steel-users.' i2mo,  2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. Svo,  5  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing Svo,  25  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  oo 

Smith's  Press-working  of  Metals Svo,  3  00 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  00 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced Svo,  5  00 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  , .  Svo,  5  00 

*  Walke's  Lectures  on  Explosives Svo,  4  00 

Ware's  Beet-sugar  Manufacture  and  Refining Small  Svo,  4  00 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover Svo,  3  00 

Wood's  Rustless  Coatings :  Corrosion  and  Electrolysis  of  Iron  and  Steel.  .Svo,  4  00 


MATHEMATICS. 

Baker's  Elliptic  Functions Svo,  i  50 

*  Bass's  Elements  of  Differential  Calculus i2mo,  4  00 

Briggs's  Elements  of  Plane  Analjrtic  Geometry i2mo,  i  00 

Compton's  Manual  of  Logarithmic  Computations i2mo,  i  50 

Davis's  Introduction  to  the  Logic  of  Algebra Svo,  i  50 

*  Dickson's  College  Algebra Large  i2mo,  i  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo,  i  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications Svo,  2  50 

Halsted's  Elements  of  Geometry Svo,  i  75 

Elementary  Synthetic  Geometry Svo,  i  50 

Rational  Geometry i2mo,  1  75 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:  Vest-pocket  size. paper,  15 

100  copies  for  5  00 

*  Mounted  on  heavy  cardboard,  8  X 10  inches,  25 

10  copies  for  2  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .Small  Svo,  3  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  Svo,  1  50 

11 


Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,     i  oo 

Johnson's  (W.  W,)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,    3  50 
Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo,     i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  00 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.) .  i2mo,    2  00 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,    3  00 

Trigonometry  and  Tables  published  separately Each,    2  00 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,    i  00 

Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward Octavo,  each     i  00 

No.  I.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
by  Robert  S.  Woodward-  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  9.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solution  of  Equations, 
byj  Mansfield  Merriman.  No.  11.  Functioas  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics Svo,    4  00 

Merriman  and  Woodward's  Higher  Mathematics Svo,    5  00 

Merriman's  Method  of  Least  Squares Svo,    2  00 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  Svo,    3  00 

Differential  and  Integral  Calculus.     2  vols,  in  one Small  Svo,    2  50 

Wood's  Elements  of  Co-ordinate  Geometry Svo,    2  00 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,    i  00 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bartlett's  Mechanical  Drawing Svo,  3  00 

*  "                  "                 "        Abridged  Ed Svo,  i  50 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  00 

Carpenter's  Experimental  Engineering Svo,  6  00 

Heating  and  Ventilating  Buildings Svo,  4  00 

Gary's  Smoke  Suppression  in  Plants  using  Bituminous  CoaL     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  Svo,  4  00 

Coolidge's  Manual  of  Drawing ,  .  .  .Svo,  paper,  i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Durley's  Kinematics  of  Machines -, Svo,  4  00 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  00 

Bering's  Ready  Reference  TaUes  (Conversion  Factors) i6mo,  morocco,  2  50 

12 


Button's  The  Gas  Engine 8vo, 

Jamison's  Mechanical  Drawing 8vo, 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo, 

Part  II.     Form,  Strength,  and  Proportions  of  Parts Svo, 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco, 

Kerr's  Power  and  Power  Transmission Svo, 

Leonard's  Machine  Shop,  Tools,  and  Methods Svo, 

*  Lorenz's  Modern  Refrigerating  Machinery.  (Pope,  Haven,  and  Dean.)  .  .  Svo, 
MacCord's  Kinematics;  or,  Practical  Mechanism Svo, 

Mechanical  Drawing.- 4to, 

Velocity  Diagrams Svo, 

MacFar land's  Standard  Reduction  Factors  for  Gases Svo, 

Mahan's  Industrial  Drawing.     (Thompson.) Svo, 

Poole's  Calorific  Power  of  Fuels Svo, 

Reid's  Course  in  Mechanical  Drawing Svo, 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo, 

Richard's  Compressed  Air i2mo, 

Robinson's  Principles  of  Mechanism Svo, 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo, 

Smith's  (0.)  Press-working  of  Metals Svo, 

Smith  (A.  W.)  and  Marx's  Machine  Design Svo, 

Thurston's  Treatise  on  Friction  and  Lost  Work  in  Machinery  and  Mill 
Work Svo, 

Animal  as  a  Machine  and  I*rime  Motor,  and  the  Laws  of  Energetics.  i2mo, 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo, 

Weisbach's  Kinematics  and  the  Power  of  Transmission.  (Herrmann — 
Klein.) Svo, 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .Svo, 

Wolff's  Windmill  as  a  Prime  Mover Svo, 

Wood's  Turbines , Svo, 


MATERIALS   OP   ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset Svo,  7  50 

Church's  Mechanics  of  Engineering Svo,  6  00 

*  Greene's  Structural  Mechanics Svo,  2  50 

Johnson's  Materials  of  Construction Svo,  6  00 

Keep's  Cast  Iron Svo,  2  50 

1  anza's  AppUed  Mechanics Svo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) Svo,  7  50 

Maurer's  Technical  Mechanics Svo,  4  00 

Merriman's  Mechanics  of  Materials Svo,  5  00 

Strength  of  Materials i2mo,  i  00 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo,  2  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering 3  vols.,  Svo,  S  00 

Part  II.     Iron  and  SteeL Svo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,  2  50 

Text-book  of  the  Materials  of  Construction Svo,  5  00 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber Svo,  2  00 

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00 

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00 

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00 

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00 

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00 

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00 

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00 

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50 

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00 

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00 

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00 

2 

50 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

SteeL 8vo,  4  00 

STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram i2mo,  1  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i  50 

Dawson's  "  Engineering"  and  Electric  Traction  Pocket-book.  .  .   i6mo,  mor.,  5  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  00 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  00 

Heat  and  Heat-engines 8vo,  5  00 

Kent's  Steam  boiler  Economy ■.8vo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors   8vo,  i  00 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  s  00 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  00 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives:  Simple   Compound,  and  Electric. i2mo,  250 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  00 

Sinclair's  Locomotive  Engine  Running  and  Management .i2mo,  2  00 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

Snow's  Steam-boiler  Practice 8vo,  3  00 

Spangler's  Valve-gears 8vo,  2  so 

Notes  on  Thermodynamics i2mo,  i  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  00 

Thurston's  Handy  Tables 8vo,  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  00 

Part  I.     History,  Structure,  and  Theory 8vo,  6  00 

Part  II.     Design,  Construction,  and  Operation 8vo,  6  00 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  00 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo,  5  00 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  00 

Whitham's  Steam-engine  Design 8vo,  5  00 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines. .  .8vo,  4  00 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Notes  and  Examples  in  Mechanics 8vo,  2  00 

Compton's  First  Lessons  in  Metal-working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  i  50 

14 


Cromwell's  Treatise  on  Toothed  Gearing. . , i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo>  -  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .i2mo,  i  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  00 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  00 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics Svo,  4  00 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  5© 

VoL  n Small  4to,  10  00 

Durley's  Kinematics  of  Machines Svo,  4  00 

Fitzgerald's  Boston  Machinist i6mo,  i  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Goss's  Locomotive  Sparks Svo,  2  00 

*  Greene's  Structural  Mechanics Svo,  2  50 

Hall's  Car  Lubrication i2mo,  i  00 

Holly's  Art  of  Saw  Filing iSmo,  7S 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  Svo,  2  00 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  00 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods Svo,  2  00 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery Svo,  i  50 

Part  11.     Form,  Strength,  and  Proportions  of  Parts Svo,  3  00 

Kerr's  Power  and  Power  Transmission Svo,  2  00 

Lanza's  Applied  Mechanics Svo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods Svo,  4  00 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.). Svo,  4  00 
MacCord'*  Kinematics;  or.  Practical  Mechanism Svo,  5  00 

Velocity  Diagrams Svo,  i  50 

Maurer's  Technical  Mechanics Svo,  4  00 

Merriman's  Mechanics  of  Materials Svo,  5  00 

*  Elements  of  Mechanics .,. .* i2mo,  i  00 

*  Michie's  Elements  of  Analytical  Mechanics Svd,  4  00 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing Svo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo,  3  00 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism Svo,  3  00 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     VoL  I Svo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo,  3  00 

Sinclair's  Locomotive-engine  Running  and  Management. i2mo,  2  00 

Smith's  (0.)  Press-working  of  Metals Svo,  3  00 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  00 

Smith  (A.  W.)  and  Marx's  Machine  Design Svo,  3  00 

Spangler,  Greent.and  Marshah's  Elements  of  Steam-engineering Svo,  3  00 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work Svo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  I  00 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   ( Herrmann — Klein. ) .  Svo ,  500 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.). Svo,  5  00 

Wood's  Elements  of  Analytical  Mechanics Svo,  3  00 

Principles  of  Elementary  Mechanics i2mo,  1  25 

Turbines Svo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

15 


METALLURGY. 


Egleston's  Metallurgy  ©f  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

Vol.  II.     Gold  and  Mercury 8vo,  7  50 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.)i2mo.  3  00 

Metcalf' s  Steel.     A  Manual  for  Steel-users i2mo,  2  00 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). . .  .  i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Smith's  Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

Part    II.     Iron  and  Steel 8vo,  3  so 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  00 


MINERALOGY. 


Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco, 

Boyd's  Resources  of  Southwest  Virginia 8vo, 

Map  of  Southwest  Virignia Pocket-book  form. 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo, 

Chester's  Catalogue  of  Minerals 8vo,  paper. 

Cloth, 

Dictionary  of  the  Names  of  Minerals 8vo, 

Dana's  System  of  Mineralogy Large  8vo,  half  leather. 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  Svo, 

Text-book  of  Mineralogy.  .*. 8vo, 

Minerals  and  How  to  Study  Them i2mo, 

Catalogue  of  American  Localities  of  Minerals Large  Svo, 

Manual  of  Mineralogy  and  Petrography i2mo, 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo, 

Eakle's  Mineral  Tables Svo, 

Egleston's  Catalogue  of  Minerals  and  Synonyms Svo, 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.). Small  Svo, 
Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses Svo, 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo,  paper,        50 
Rosenbusch's   Microscopical   Physiography   of   the   Rock-making  Minerals. 

(Iddings.) Svo,    5  00 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks Svo,    2  00 


MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia Svo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  00 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to,hf.  mor.,  25  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

16 


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I 

00 

I 

25 

2 

50 

2 

00 

4 

00 

Fowler's  Sewage  Works  Analyses i2mo,  2  00 

Goodyear 's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo,  2  50 

Ihlseng's  Manual  of  Mining 8vo,  5  00 

**  lles's  Lead-smelting.     (Postage  gc.  additionaL) i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

O'Driscoll's  Notes  on  the  "treatment  of  Gold  Ores 8vo,  2  00 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) Svo, 

*  Walke's  Lectures  on  Explosives Svo,  4  00 

Wilson's  Cyanide  Processes I2m0t  i  50 

Chlorination  Process i2mo,  i  50 

Hydraulic  and  Placer  Mining i2mo,  2  00 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo,  i  25 


SANITARY  SCIENCE. 

Bashore's  Sanitation  of  a  Country  House X2mo,  1  00 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) Svo,  3  00 

Water-supply  Engineering Svo,  4  00 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  00 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  Svo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies Svo,  3  00 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Svo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  Svo,  4  00 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Ogden's  Sewer  Design i2mo,  2  00 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

*  Price's  Handbook  on  Sanitation i2mo,  j  50 

Richards's  Cost  of  Food.     A  Study  in  bietaries i2mo,  i  00 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  00 

Richards  and  Woodman's  Air.  W.ater,  and  Food  from  a  Sanitary  Stand- 
point  Svo,  2  00 

*  Richards  and  Williams's  The  Dietary  Computer Svo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage Svo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies Svo,  s  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  00 

Whipple's  Microscopy  of  Drinking-water Svo,  3  50 

Winton's  Microscopy  of  Vegetable  Foods Svo,  7  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  x  50 


MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoflf  and  Collins.). . .  .Large  i2mo,  3  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  Svo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds Svo.  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Fallacy  of  the  Present  Theory  of  Sound i6mo,  x  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894.. Small  Svo,  3  00 

Rostoski's.Serum  Diagnosis.     (Bolduan.) i2mo,  i  00 

Rotherham's  Emphasized  New  Testament Large  Svo,  2  00 

17 


Steel's  Treatise  on  the  Diseases  of  the  Dog .8vo,  3  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  00 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital .  lamo,  i  25 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 


Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  00 

Oesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  s  00 

Letteris's  Hebrew  Bible 8vo,  2  25 

18 


7 


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THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $I.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


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